Right Triangles and Trigonometry

Transcription

1 Right Triangles and Trigonometry The Pythagorean Theorem Special Right Triangles Similar Right Triangles The Tangent Ratio The Sine and osine Ratios Solving Right Triangles Footbridge Footb bridge d ((p. p. 6 6) )) (p. ) Skiing (p SEE the ig Idea Washington Monument (p (p. ) Rock Wall (p. (p ) Fi Escape Fire Escape (p. (p. ) ) int_math_pe_09co.indd /0/ : M

2 Maintaining Mathematical Proficiency Using Properties of Radicals Eample Simplify 8. 8 = 6 Eample Simplify. Factor using the greatest perfect square factor. = 6 Product Property of Square Roots = 8 Simplify. = = = Multiply by. Product Property of Square Roots Simplify. Simplify the epression Solving Proportions Eample Solve 0 =. 0 = Write the proportion. = 0 ross Products Property = 0 Multiply. = 0 ivide each side by. = Simplify. Solve the proportion. 7. = 8. = 9. = =. + =. 9 =. STRT RESONING The Product Property of Square Roots allows you to simplify the square root of a product. re you able to simplify the square root of a sum? of a difference? Eplain. ynamic Solutions available at igideasmath.com

3 Mathematical Practices ttending to Precision ore oncept Standard Position for a Right Triangle In unit circle trigonometry, a right triangle is in standard position when:. The hypotenuse is a radius of the circle of radius with center at the origin.. One leg of the right triangle lies on the -ais.. The other leg of the right triangle is perpendicular to the -ais. Mathematically profi cient students epress numerical answers precisely y rawing an Isosceles Right Triangle in Standard Position Use dynamic geometry software to construct an isosceles right triangle in standard position. What are the eact coordinates of its vertices? Sample Points (0, 0) (0.7, 0.7) (0.7, 0) Segments = = 0.7 = 0.7 ngle m = To determine the eact coordinates of the vertices, label the length of each leg. y the Pythagorean Theorem, which you will study in Section 9., + =. Solving this equation yields =, or. So, the eact coordinates of the vertices are (0, 0), (, Monitoring Progress ), and ( ), 0.. Use dynamic geometry software to construct a right triangle with acute angle measures of 0 and 60 in standard position. What are the eact coordinates of its vertices?. Use dynamic geometry software to construct a right triangle with acute angle measures of 0 and 70 in standard position. What are the approimate coordinates of its vertices? 6 hapter 9 Right Triangles and Trigonometry

4 9. The Pythagorean Theorem Essential Question How can you prove the Pythagorean Theorem? Proving the Pythagorean Theorem without Words Work with a partner. a. raw and cut out a right triangle with legs a and b, and hypotenuse c. b. Make three copies of your right triangle. rrange all four triangles to form a large square, as shown. c. Find the area of the large square in terms of a, b, and c by summing the areas of the triangles and the small square. b a a c c b b c c a a b d. opy the large square. ivide it into two smaller squares and two equally-sized rectangles, as shown. a b e. Find the area of the large square in terms of a and b by summing the areas of the rectangles and the smaller squares. b b f. ompare your answers to parts (c) and (e). Eplain how this proves the Pythagorean Theorem. a a b a Proving the Pythagorean Theorem Work with a partner. a. raw a right triangle with legs a and b, and hypotenuse c, as shown. raw the altitude from to. Label the lengths, as shown. RESONING STRTLY To be proficient in math, you need to know and fleibly use different properties of operations and objects. b h c d c b. Eplain why,, and are similar. d a c. Write a two-column proof using the similar triangles in part (b) to prove that a + b = c. ommunicate Your nswer. How can you prove the Pythagorean Theorem?. Use the Internet or some other resource to find a way to prove the Pythagorean Theorem that is different from Eplorations and. Section 9. The Pythagorean Theorem 7

5 9. Lesson What You Will Learn ore Vocabulary Pythagorean triple, p. 8 Previous right triangle legs of a right triangle hypotenuse Use the Pythagorean Theorem. Use the onverse of the Pythagorean Theorem. lassify triangles. Using the Pythagorean Theorem One of the most famous theorems in mathematics is the Pythagorean Theorem, named for the ancient Greek mathematician Pythagoras. This theorem describes the relationship between the side lengths of a right triangle. Theorem Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Proof Eplorations and, p. 7; E. 9, p. 8 a c b c = a + b STUY TIP You may find it helpful to memorize the basic Pythagorean triples, shown in bold, for standardized tests. Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c = a + b. ore oncept ommon Pythagorean Triples and Some of Their Multiples,, 6, 8, 0 9,,,,,, 0,, 6, 6, 9,, 8,, 7 6, 0,,, 8,, 7 7,,, 8, 0, 7, 7 7,, The most common Pythagorean triples are in bold. The other triples are the result of multiplying each integer in a bold-faced triple by the same factor. Using the Pythagorean Theorem Find the value of. Then tell whether the side lengths form a Pythagorean triple. c = a + b = + = + = 69 = Pythagorean Theorem Substitute. Multiply. dd. Find the positive square root. The value of is. ecause the side lengths,, and are integers that satisfy the equation c = a + b, they form a Pythagorean triple. 8 hapter 9 Right Triangles and Trigonometry

6 Using the Pythagorean Theorem Find the value of. Then tell whether the side lengths form a Pythagorean triple. 7 ONNETIONS TO LGER The Product Property of Square Roots on page 9 states that the square root of a product equals the product of the square roots of the factors. c = a + b Pythagorean Theorem = 7 + Substitute. 96 = 9 + Multiply. 7 = Subtract 9 from each side. 7 = Find the positive square root. 9 = Product Property of Square Roots 7 = Simplify. The value of is 7. ecause 7 is not an integer, the side lengths do not form a Pythagorean triple. Solving a Real-Life Problem The skyscrapers shown are connected by a skywalk with support beams. Use the Pythagorean Theorem to approimate the length of each support beam. 7.7 m Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. = (.6) + (7.7) Pythagorean Theorem = (.6) + (7.7) Find the positive square root..9 Use a calculator to approimate. The length of each support beam is about.9 meters..6 m support beams 7.7 m Monitoring Progress Help in English and Spanish at igideasmath.com Find the value of. Then tell whether the side lengths form a Pythagorean triple n anemometer is a device used to measure wind speed. The anemometer shown is attached to the top of a pole. Support wires are attached to the pole feet above the ground. Each support wire is 6 feet long. How far from the base of the pole is each wire attached to the ground? 6 ft d ft Section 9. The Pythagorean Theorem 9

7 Using the onverse of the Pythagorean Theorem The converse of the Pythagorean Theorem is also true. You can use it to determine whether a triangle with given side lengths is a right triangle. Theorem onverse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c = a + b, then is a right triangle. a c b Proof E. 9, p. Verifying Right Triangles Tell whether each triangle is a right triangle. USING TOOLS STRTEGILLY Use a calculator to determine that 0.60 is the length of the longest side in part (a). a. 8 7 b. 9 Let c represent the length of the longest side of the triangle. heck to see whether the side lengths satisfy the equation c = a + b. a. ( ) =? =? = The triangle is a right triangle. b. ( 9 ) =? + 6 ( 9 ) =? =? The triangle is not a right triangle. Monitoring Progress Help in English and Spanish at igideasmath.com Tell whether the triangle is a right triangle hapter 9 Right Triangles and Trigonometry

8 lassifying Triangles The onverse of the Pythagorean Theorem is used to determine whether a triangle is a right triangle. You can use the theorem below to determine whether a triangle is acute or obtuse. Theorem Pythagorean Inequalities Theorem For any, where c is the length of the longest side, the following statements are true. If c < a + b, then is acute. If c > a + b, then is obtuse. b c b c REMEMER The Triangle Inequality Theorem on page 8 states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. a c < a + b Proof Es. and, p. lassifying Triangles Verify that segments with lengths of. feet,. feet, and 6. feet form a triangle. Is the triangle acute, right, or obtuse? a c > a + b Step Use the Triangle Inequality Theorem to verify that the segments form a triangle.. +. >? >? >?. 9. > >.. >. The segments with lengths of. feet,. feet, and 6. feet form a triangle. Step lassify the triangle by comparing the square of the length of the longest side with the sum of the squares of the lengths of the other two sides. c a + b ompare c with a + b Substitute Simplify. 7. <. c is less than a + b. The segments with lengths of. feet,. feet, and 6. feet form an acute triangle. Monitoring Progress Help in English and Spanish at igideasmath.com 6. Verify that segments with lengths of,, and 6 form a triangle. Is the triangle acute, right, or obtuse? 7. Verify that segments with lengths of.,.8, and. form a triangle. Is the triangle acute, right, or obtuse? Section 9. The Pythagorean Theorem

9 9. Eercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. VOULRY What is a Pythagorean triple?. IFFERENT WORS, SME QUESTION Which is different? Find both answers. Find the length of the longest side. Find the length of the hypotenuse. Find the length of the longest leg. Find the length of the side opposite the right angle. Monitoring Progress and Modeling with Mathematics In Eercises 6, find the value of. Then tell whether the side lengths form a Pythagorean triple. (See Eample.) In Eercises 7 0, find the value of. Then tell whether the side lengths form a Pythagorean triple. (See Eample.) ERROR NLYSIS In Eercises and, describe and correct the error in using the Pythagorean Theorem c = a + b = 7 + = (7 + ) = = c = a + b = = = 776 = hapter 9 Right Triangles and Trigonometry

10 . MOELING WITH MTHEMTIS The fire escape forms a right triangle, as shown. Use the Pythagorean Theorem to approimate the distance between the two platforms. (See Eample.) In Eercises 8, verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse? (See Eample.). 0,, and. 6, 8, and ft., 6, and 0., 0, and 6.., 6.7, and , 8., and. 7., 0, and ,, and 8.9 ft. MOELING WITH MTHEMTIS The backboard of the basketball hoop forms a right triangle with the supporting rods, as shown. Use the Pythagorean Theorem to approimate the distance between the rods where they meet the backboard. 9. MOELING WITH MTHEMTIS In baseball, the lengths of the paths between consecutive bases are 90 feet, and the paths form right angles. The player on first base tries to steal second base. How far does the ball need to travel from home plate to second base to get the player out? 0. RESONING You are making a canvas frame for a painting using stretcher bars. The rectangular painting will be 0 inches long and 8 inches wide. Using a ruler, how can you be certain that the corners of the frame are 90?. in. 9.8 in. In Eercises 0, tell whether the triangle is a right triangle. (See Eample.) In Eercises, find the area of the isosceles triangle... 7 m 0 cm h 6 m h 7 m 0 cm.. 0 ft h ft 0 ft cm 0 m h 0 m m Section 9. The Pythagorean Theorem

11 . NLYZING RELTIONSHIPS Justify the istance Formula using the Pythagorean Theorem. 6. HOW O YOU SEE IT? How do you know is a right angle? PROLEM SOLVING You are making a kite and need to figure out how much binding to buy. You need the binding for the perimeter of the kite. in. The binding comes in packages of two yards. How many packages should you buy? 8 in. 0 in. in. 8. PROVING THEOREM Use the Pythagorean Theorem to prove the Hypotenuse-Leg (HL) ongruence Theorem. 9. PROVING THEOREM Prove the onverse of the Pythagorean Theorem. (Hint: raw with side lengths a, b, and c, where c is the length of the longest side. Then draw a right triangle with side lengths a, b, and, where is the length of the hypotenuse. ompare lengths c and.) 0. THOUGHT PROVOKING onsider two positive integers m and n, where m > n. o the following epressions produce a Pythagorean triple? If yes, prove your answer. If no, give a countereample. mn, m n, m + n. MKING N RGUMENT Your friend claims 7 and 7 cannot be part of a Pythagorean triple because does not equal a positive integer squared. Is your friend correct? Eplain your reasoning.. PROVING THEOREM opy and complete the proof of the Pythagorean Inequalities Theorem when c < a + b. Given In, c < a + b, where c is the length of the longest side. PQR has side lengths a, b, and, where is the length of the hypotenuse, and R is a right angle. Prove is an acute triangle. STTEMENTS c b a Q a R. In, c < a + b, where c is the length of the longest side. PQR has side lengths a, b, and, where is the length of the hypotenuse, and R is a right angle. P b RESONS.. a + b =.. c <.. c <. Take the positive square root of each side.. m R = m < m R 6. onverse of the Hinge Theorem 7. m < is an acute angle is an acute triangle. 9.. PROVING THEOREM Prove the Pythagorean Inequalities Theorem when c > a + b. (Hint: Look back at Eercise.) Maintaining Mathematical Proficiency Simplify the epression by rationalizing the denominator. (Section.) Reviewing what you learned in previous grades and lessons 8 7. hapter 9 Right Triangles and Trigonometry

12 9. Special Right Triangles Essential Question What is the relationship among the side lengths of triangles? triangles? Side Ratios of an Isosceles Right Triangle TTENING TO PREISION To be proficient in math, you need to epress numerical answers with a degree of precision appropriate for the problem contet. Work with a partner. a. Use dynamic geometry software to construct an isosceles right triangle with a leg length of units. b. Find the acute angle measures. Eplain why this triangle is called a triangle. c. Find the eact ratios of the side lengths (using square roots). = = = 0 0 d. Repeat parts (a) and (c) for several other isosceles right triangles. Use your results to write a conjecture about the ratios of the side lengths of an isosceles right triangle. Sample Points (0, ) (, 0) (0, 0) Segments =.66 = = ngles m = m = Side Ratios of a Triangle Work with a partner. a. Use dynamic geometry software to construct a right triangle with acute angle measures of 0 and 60 (a triangle), where the shorter leg length is units. b. Find the eact ratios Sample of the side lengths (using square roots). = = = 0 0 c. Repeat parts (a) and (b) for several other triangles. Use your results to write a conjecture about the ratios of the side lengths of a triangle. ommunicate Your nswer. What is the relationship among the side lengths of triangles? triangles? Points (0,.0) (, 0) (0, 0) Segments = 6 = =.0 ngles m = 0 m = 60 Section 9. Special Right Triangles

13 9. Lesson What You Will Learn ore Vocabulary Previous isosceles triangle Find side lengths in special right triangles. Solve real-life problems involving special right triangles. Finding Side Lengths in Special Right Triangles triangle is an isosceles right triangle that can be formed by cutting a square in half diagonally. REMEMER n epression involving a radical with inde is in simplest form when no radicands have perfect squares as factors other than, no radicands contain fractions, and no radicals appear in the denominator of a fraction. Finding Side Lengths in Triangles Find the value of. Write your answer in simplest form. a. Theorem Triangle Theorem In a triangle, the hypotenuse is times as long as each leg. 8 b. Proof E. 9, p. 0 hypotenuse = leg a. y the Triangle Sum Theorem, the measure of the third angle must be, so the triangle is a triangle. hypotenuse = leg = 8 = 8 The value of is Triangle Theorem Substitute. Simplify. b. y the ase ngles Theorem and the orollary to the Triangle Sum Theorem, the triangle is a triangle. hypotenuse = leg = Substitute Triangle Theorem = ivide each side by. = Simplify. The value of is. 6 hapter 9 Right Triangles and Trigonometry

14 Theorem Triangle Theorem In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. Proof E., p hypotenuse = shorter leg longer leg = shorter leg REMEMER ecause the angle opposite 9 is larger than the angle opposite, the leg with length 9 is longer than the leg with length by the Triangle Larger ngle Theorem. Finding Side Lengths in a Triangle Find the values of and y. Write your answer in simplest form. Step Find the value of. longer leg = shorter leg Triangle Theorem 9 = Substitute. 9 = ivide each side by. 9 = Multiply by. 9 = Multiply fractions. = Simplify. y The value of is. Step Find the value of y. hypotenuse = shorter leg y = y = Triangle Theorem Substitute. Simplify. The value of y is 6. Monitoring Progress Help in English and Spanish at igideasmath.com Find the value of the variable. Write your answer in simplest form... y h Section 9. Special Right Triangles 7

15 Solving Real-Life Problems Modeling with Mathematics The road sign is shaped like an equilateral triangle. Estimate the area of the sign by finding the area of the equilateral triangle. First find the height h of the triangle by dividing it into two triangles. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 8 inches. h = 8 = Triangle Theorem Use h = 8 to find the area of the equilateral triangle. rea = bh = (6) ( 8 ) 6.8 The area of the sign is about 6 square inches. 6 in. YIEL 8 in. 8 in h 6 in. 6 in. Finding the Height of a Ramp tipping platform is a ramp used to unload trucks. How high is the end of an 80-foot ramp when the tipping angle is 0?? ramp 80 ft height of ramp tipping angle ft 60 When the tipping angle is 0, the height h of the ramp is the length of the shorter leg of a triangle. The length of the hypotenuse is 80 feet. 80 = h Triangle Theorem 0 = h ivide each side by. When the tipping angle is, the height h of the ramp is the length of a leg of a triangle. The length of the hypotenuse is 80 feet. 80 = h Triangle Theorem 80 = h ivide each side by. 6.6 h Use a calculator. When the tipping angle is 0, the ramp height is 0 feet. When the tipping angle is, the ramp height is about 6 feet 7 inches. Monitoring Progress Help in English and Spanish at igideasmath.com. The logo on a recycling bin resembles an equilateral triangle with side lengths of 6 centimeters. pproimate the area of the logo. 6. The body of a dump truck is raised to empty a load of sand. How high is the -foot-long body from the frame when it is tipped upward by a 60 angle? 8 hapter 9 Right Triangles and Trigonometry

16 9. Eercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. VOULRY Name two special right triangles by their angle measures.. WRITING Eplain why the acute angles in an isosceles right triangle always measure. Monitoring Progress and Modeling with Mathematics In Eercises 6, find the value of. Write your answer in simplest form. (See Eample.) In Eercises 7 0, find the values of and y. Write your answers in simplest form. (See Eample.) y y ERROR NLYSIS In Eercises and, describe and correct the error in finding the length of the hypotenuse y the Triangle Sum Theorem, the measure of the third angle must be 60. So, the triangle is a triangle. hypotenuse = shorter leg = 7 y y 0. y the Triangle Sum Theorem, the measure of the third angle must be. So, the triangle is a triangle. hypotenuse = leg leg = So, the length of the hypotenuse is units. In Eercises and, sketch the figure that is described. Find the indicated length. Round decimal answers to the nearest tenth.. The side length of an equilateral triangle is centimeters. Find the length of an altitude.. The perimeter of a square is 6 inches. Find the length of a diagonal. In Eercises and 6, find the area of the figure. Round decimal answers to the nearest tenth. (See Eample.). 8 ft 6. m 60 m 7. PROLEM SOLVING Each half of the drawbridge is about 8 feet long. How high does the drawbridge rise when is 0?? 60? (See Eample.) 8 ft m m So, the length of the hypotenuse is 7 units. Section 9. Special Right Triangles 9

17 8. MOELING WITH MTHEMTIS nut is shaped like a regular heagon with side lengths of centimeter. Find the value of. (Hint: regular heagon can be divided into si congruent triangles.) cm. THOUGHT PROVOKING The diagram below is called the illes rectangle. Each triangle in the diagram has rational angle measures and each side length contains at most one square root. Label the sides and angles in the diagram. escribe the triangles. 9. PROVING THEOREM Write a paragraph proof of the Triangle Theorem. Given EF is a triangle. Prove The hypotenuse is times as long as each leg. F E 0. HOW O YOU SEE IT? The diagram shows part of the Wheel of Theodorus. 60. WRITING escribe two ways to show that all isosceles right triangles are similar to each other.. MKING N RGUMENT Each triangle in the diagram is a triangle. t Stage 0, the legs of the triangle are each unit long. Your brother claims the lengths of the legs of the triangles added are halved at each stage. So, the length of a leg of a triangle added in Stage 8 will be unit. Is your 6 brother correct? Eplain your reasoning. 6 7 Stage 0 Stage Stage a. Which triangles, if any, are triangles? b. Which triangles, if any, are triangles?. PROVING THEOREM Write a paragraph proof of the Triangle Theorem. (Hint: onstruct JML congruent to JKL.) Given JKL is a triangle. K Prove The hypotenuse is twice 60 as long as the shorter 0 J leg, and the longer leg L is times as long as the shorter leg. M Maintaining Mathematical Proficiency Find the value of. (Section 8.) 6. EF LMN 7. QRS Stage Stage. USING STRUTURE TUV is a triangle, where two vertices are U(, ) and V(, ), UV is the hypotenuse, and point T is in Quadrant I. Find the coordinates of T. Reviewing what you learned in previous grades and lessons E N 0 F 0 L M. 7 S R Q 0 hapter 9 Right Triangles and Trigonometry

18 9. Similar Right Triangles Essential Question How are altitudes and geometric means of right triangles related? Writing a onjecture Work with a partner. a. Use dynamic geometry software to construct right, as shown. raw so that it is an altitude from the right angle to the hypotenuse of Points (0, ) (8, 0) (0, 0) (.,.6) Segments = 9. = 8 = ONSTRUTING VILE RGUMENTS To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results in constructing arguments. b. The geometric mean of two positive numbers a and b is the positive number that satisfies a =. is the geometric mean of a and b. b Write a proportion involving the side lengths of and so that is the geometric mean of two of the other side lengths. Use similar triangles to justify your steps. c. Use the proportion you wrote in part (b) to find. d. Generalize the proportion you wrote in part (b). Then write a conjecture about how the geometric mean is related to the altitude from the right angle to the hypotenuse of a right triangle. Work with a partner. Use a spreadsheet to find the arithmetic mean and the geometric mean of several pairs of positive numbers. ompare the two means. What do you notice? ommunicate Your nswer omparing Geometric and rithmetic Means a b rithmetic Mean Geometric Mean How are altitudes and geometric means of right triangles related? Section 9. Similar Right Triangles

19 9. Lesson What You Will Learn ore Vocabulary geometric mean, p. Previous altitude of a triangle similar figures Identify similar triangles. Solve real-life problems involving similar triangles. Use geometric means. Identifying Similar Triangles When the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other. Theorem Right Triangle Similarity Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.,, and. Proof E., p. 8 Identifying Similar Triangles Identify the similar triangles in the diagram. U S R T Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. T S S T U R U R T TSU RTU RST Monitoring Progress Help in English and Spanish at igideasmath.com Identify the similar triangles.. Q. E H F T S R G hapter 9 Right Triangles and Trigonometry

20 Solving Real-Life Problems Modeling with Mathematics roof has a cross section that is a right triangle. The diagram shows the approimate dimensions of this cross section. Find the height h of the roof. Y. m h. m Z 6. m W X. Understand the Problem You are given the side lengths of a right triangle. You need to find the height of the roof, which is the altitude drawn to the hypotenuse.. Make a Plan Identify any similar triangles. Then use the similar triangles to write a proportion involving the height and solve for h.. Solve the Problem Identify the similar triangles and sketch them. Z Z OMMON ERROR Notice that if you tried to write a proportion using XYW and YZW, then there would be two unknowns, so you would not be able to solve for h.. m X Y h W. m Y h W X 6. m XYW YZW XZY. m Y. m ecause XYW XZY, you can write a proportion. YW ZY = XY orresponding side lengths of similar triangles are proportional. XZ h. =. Substitute. 6. h.7 Multiply each side by.. The height of the roof is about.7 meters.. Look ack ecause the height of the roof is a leg of right YZW and right XYW, it should be shorter than each of their hypotenuses. The lengths of the two hypotenuses are YZ =. and XY =.. ecause.7 <., the answer seems reasonable. Monitoring Progress Find the value of. Help in English and Spanish at igideasmath.com. E G H F. J K L M Section 9. Similar Right Triangles

21 Using a Geometric Mean ore oncept Geometric Mean The geometric mean of two positive numbers a and b is the positive number that satisfies a = b. So, = ab and = ab. Finding a Geometric Mean Find the geometric mean of and 8. = ab efinition of geometric mean = 8 Substitute for a and 8 for b. = 8 Take the positive square root of each side. = Factor. = Simplify. The geometric mean of and 8 is.9. In right, altitude is drawn to the hypotenuse, forming two smaller right triangles that are similar to. From the Right Triangle Similarity Theorem, you know that. ecause the triangles are similar, you can write and simplify the following proportions involving geometric means. = = = = = = Theorems Geometric Mean (ltitude) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse. Proof E., p. 8 = Geometric Mean (Leg) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. = = Proof E., p. 8 hapter 9 Right Triangles and Trigonometry

22 Using a Geometric Mean OMMON ERROR In Eample (b), the Geometric Mean (Leg) Theorem gives y = ( + ), not y = ( + ), because the side with length y is adjacent to the segment with length. Find the value of each variable. a. 6 b. y a. pply the Geometric Mean b. pply the Geometric Mean (ltitude) Theorem. (Leg) Theorem. = 6 y = ( + ) = 8 y = 7 = 8 y = = 9 y = = The value of y is. The value of is. Using Indirect Measurement To find the cost of installing a rock wall in your school gymnasium, you need to find the height of the gym wall. You use a cardboard square to line up the top and bottom of the gym wall. Your friend measures the vertical distance from the ground to your eye and the horizontal distance from you to the gym wall. pproimate the height of the gym wall. y the Geometric Mean (ltitude) Theorem, you know that 8. is the geometric mean of w and. 8. = w Geometric Mean (ltitude) Theorem 7. = w Square 8... = w ivide each side by. The height of the wall is + w = +. = 9. feet. 8. ft w ft ft Monitoring Progress Help in English and Spanish at igideasmath.com Find the geometric mean of the two numbers.. and and 7. 6 and 8 8. Find the value of in the triangle at the left WHT IF? In Eample, the vertical distance from the ground to your eye is. feet and the distance from you to the gym wall is 9 feet. pproimate the height of the gym wall. Section 9. Similar Right Triangles

23 9. Eercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. OMPLETE THE SENTENE If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and.. WRITING In your own words, eplain geometric mean. Monitoring Progress and Modeling with Mathematics In Eercises and, identify the similar triangles. (See Eample.). F E In Eercises 8, find the geometric mean of the two numbers. (See Eample.). 8 and. 9 and 6 H G. and 0. and. 6 and 6. 8 and 8. M 7. 7 and 6 8. and L N K In Eercises 0, find the value of. (See Eample.). 6. Q W Y T 0 7 X Z In Eercises 9 6, find the value of the variable. (See Eample.) y 8 0 S 6 R y E H F G 0 b ft.8 ft.8 ft.6 ft. ft. z ft 6 hapter 9 Right Triangles and Trigonometry

24 ERROR NLYSIS In Eercises 7 and 8, describe and correct the error in writing an equation for the given diagram. 7. y z MTHEMTIL ONNETIONS In Eercises, find the value(s) of the variable(s).. a b + 8. w v z = w (w + v) e g f d. y z 6.. RESONING Use the diagram. ecide which proportions are true. Select all that apply. z y h d = f h MOELING WITH MTHEMTIS In Eercises 9 and 0, use the diagram. (See Eample.) = = = = 7. ft. ft 6 ft 9. ft E. 9 E You want to determine the height of a monument at a local park. You use a cardboard square to line up the top and bottom of the monument, as shown at the above left. Your friend measures the vertical distance from the ground to your eye and the horizontal distance from you to the monument. pproimate the height of the monument. 0. Your classmate is standing on the other side of the monument. She has a piece of rope staked at the base of the monument. She etends the rope to the cardboard square she is holding lined up to the top and bottom of the monument. Use the information in the diagram above to approimate the height of the monument. o you get the same answer as in Eercise 9? Eplain your reasoning. 6. NLYZING RELTIONSHIPS You are designing a diamond-shaped kite. You know that =.8 centimeters, = 7 centimeters, and = 8.8 centimeters. You want to use a straight crossbar. bout how long should it be? Eplain your reasoning. 7. NLYZING RELTIONSHIPS Use both of the Geometric Mean Theorems to find and. 0 Section 9. Similar Right Triangles 7

25 8. HOW O YOU SEE IT? In which of the following triangles does the Geometric Mean (ltitude) Theorem apply? 0. MKING N RGUMENT Your friend claims the geometric mean of and 9 is 6, and then labels the triangle, as shown. Is your friend correct? Eplain 9 your reasoning. 6 In Eercises and, use the given statements to prove the theorem. Given is a right triangle. ltitude is drawn to hypotenuse.. PROVING THEOREM Prove the Geometric Mean (ltitude) Theorem by showing that =. 9. PROVING THEOREM Use the diagram of. opy and complete the proof of the Pythagorean Theorem. Given In, is a right angle. Prove c = a + b STTEMENTS. In, is a right angle.. raw a perpendicular segment (altitude) from to. RESONS.. Perpendicular Postulate. ce = a and cf = b.. ce + b = + b. ddition Property of Equality. ce + cf = a + b. 6. c(e + f ) = a + b e + f = 7. Segment ddition Postulate 8. c c = a + b c = a + b 9. Simplify. b f a c e. PROVING THEOREM Prove the Geometric Mean (Leg) Theorem by showing that = and =.. RITIL THINKING raw a right isosceles triangle and label the two leg lengths. Then draw the altitude to the hypotenuse and label its length y. Now, use the Right Triangle Similarity Theorem to draw the three similar triangles from the image and label any side length that is equal to either or y. What can you conclude about the relationship between the two smaller triangles? Eplain your reasoning.. THOUGHT PROVOKING The arithmetic mean and geometric mean of two nonnegative numbers and y are shown. arithmetic mean = + y geometric mean = y Write an inequality that relates these two means. Justify your answer.. PROVING THEOREM Prove the Right Triangle Similarity Theorem by proving three similarity statements. Given is a right triangle. ltitude is drawn to hypotenuse. Prove,, Maintaining Mathematical Proficiency Solve the equation for. (Skills Review Handbook) 6. = 7. 9 = Reviewing what you learned in previous grades and lessons 8. 9 = = 8 hapter 9 Right Triangles and Trigonometry

26 9. 9. What id You Learn? ore Vocabulary Pythagorean triple, p. 8 geometric mean, p. ore oncepts Section 9. Pythagorean Theorem, p. 8 ommon Pythagorean Triples and Some of Their Multiples, p. 8 onverse of the Pythagorean Theorem, p. 0 Pythagorean Inequalities Theorem, p. Section Triangle Theorem, p Triangle Theorem, p. 7 Section 9. Right Triangle Similarity Theorem, p. Geometric Mean (ltitude) Theorem, p. Geometric Mean (Leg) Theorem, p. Mathematical Practices. In Eercise on page, describe the steps you took to find the area of the triangle.. In Eercise on page 0, can one of the ways be used to show that all triangles are similar? Eplain.. Eplain why the Geometric Mean (ltitude) Theorem does not apply to three of the triangles in Eercise 8 on page 8. Form a Weekly Study Group, Set Up Rules onsider using the following rules. Members must attend regularly, be on time, and participate. The sessions will focus on the key math concepts, not on the needs of one student. Students who skip classes will not be allowed to participate in the study group. Students who keep the group from being productive will be asked to leave the group. 9

27 9. 9. Quiz Find the value of. Tell whether the side lengths form a Pythagorean triple. (Section 9.) Verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse? (Section 9.).,, and 0. 7, 9, and 6.,, and 0 Find the values of and y. Write your answers in simplest form. (Section 9.) 7. 6 y 8. 8 y y 60 Find the geometric mean of the two numbers. (Section 9.) 0. 6 and. and 0. 8 and 6 Identify the similar right triangles. Then find the value of the variable. (Section 9.). 8. E F 6 H y 9 G. J 8 z M K L 6. Television sizes are measured by the length of their diagonal. You want to purchase a television that is at least 0 inches. Should you purchase the television shown? Eplain your reasoning. (Section 9.) 7. Each triangle shown below is a right triangle. (Sections 9. 9.) a. re any of the triangles special right triangles? Eplain your reasoning. b. List all similar triangles, if any. c. Find the lengths of the altitudes of triangles and. 0. in. 6 in. 6 0 E 6 0 hapter 9 Right Triangles and Trigonometry

28 9. The Tangent Ratio Essential Question How is a right triangle used to find the tangent of an acute angle? Is there a unique right triangle that must be used? Let be a right triangle with acute. The tangent of (written as tan ) is defined as follows. length of leg opposite tan = length of leg adjacent to = adjacent opposite alculating a Tangent Ratio Work with a partner. Use dynamic geometry software. a. onstruct, as shown. onstruct segments perpendicular to to form right triangles that share verte and are similar to with vertices, as shown. 6 0 K L M N O P Q J I H G F E Sample Points (0, 0) (8, 6) (8, 0) ngle m = 6.87 TTENING TO PREISION To be proficient in math, you need to epress numerical answers with a degree of precision appropriate for the problem contet. b. alculate each given ratio to complete the table for the decimal value of tan for each right triangle. What can you conclude? Ratio tan K LE E MF F Using a alculator NG G OH H Work with a partner. Use a calculator that has a tangent key to calculate the tangent of o you get the same result as in Eploration? Eplain. PI I QJ J ommunicate Your nswer. Repeat Eploration for with vertices (0, 0), (8, ), and (8, 0). onstruct the seven perpendicular segments so that not all of them intersect at integer values of. iscuss your results.. How is a right triangle used to find the tangent of an acute angle? Is there a unique right triangle that must be used? Section 9. The Tangent Ratio

29 9. Lesson What You Will Learn ore Vocabulary trigonometric ratio, p. tangent, p. angle of elevation, p. REING Remember the following abbreviations. tangent tan opposite opp. adjacent adj. Use the tangent ratio. Solve real-life problems involving the tangent ratio. Using the Tangent Ratio trigonometric ratio is a ratio of the lengths of two sides in a right triangle. ll right triangles with a given acute angle are similar by the Similarity Theorem. So, JKL XYZ, YZ = JL. This can be rewritten as XZ KL JL = YZ, which is a trigonometric ratio. So, trigonometric XZ ratios are constant for a given angle measure. and you can write KL The tangent ratio is a trigonometric ratio for acute angles that involves the lengths of the legs of a right triangle. ore oncept Tangent Ratio Let be a right triangle with acute. The tangent of (written as tan ) is defined as follows. length of leg opposite tan = length of leg adjacent to = leg opposite J Y Z K L hypotenuse leg adjacent to X TTENING TO PREISION Unless told otherwise, you should round the values of trigonometric ratios to four decimal places and round lengths to the nearest tenth. In the right triangle above, and are complementary. So, is acute. You can use the same diagram to find the tangent of. Notice that the leg adjacent to is the leg opposite and the leg opposite is the leg adjacent to. Finding Tangent Ratios Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places. opp. S tan S = adj. to S = RT ST = 80 8 = 0 9. tan R = opp. R adj. to R = ST RT = 8 80 = 9 0 = 0.0 S 8 T 8 80 R Monitoring Progress Help in English and Spanish at igideasmath.com Find tan J and tan K. Write each answer as a fraction and as a decimal rounded to four places.. K. L J J L K hapter 9 Right Triangles and Trigonometry

30 Finding a Leg Length Find the value of. Round your answer to the nearest tenth. USING TOOLS STRTEGILLY You can also use the Table of Trigonometric Ratios available at igideasmath.com to find the decimal approimations of trigonometric ratios. Use the tangent of an acute angle to find a leg length. tan = opp. Write ratio for tangent of. adj. tan = Substitute. tan = Multiply each side by. = ivide each side by tan. tan 7.6 Use a calculator. The value of is about 7.6. STUY TIP The tangents of all 60 angles are the same constant ratio. ny right triangle with a 60 angle can be used to determine this value. You can find the tangent of an acute angle measuring 0,, or 60 by applying what you know about special right triangles. Using a Special Right Triangle to Find a Tangent Use a special right triangle to find the tangent of a 60 angle. Step ecause all triangles are similar, you can simplify your calculations by choosing as the length of the shorter leg. Use the Triangle Theorem to find the length of the longer leg. longer leg = shorter leg Triangle Theorem = Substitute. = Simplify. 60 Step Find tan 60. tan 60 = opp. adj. tan 60 = tan 60 = Write ratio for tangent of 60. Substitute. Simplify. The tangent of any 60 angle is.7. Monitoring Progress Help in English and Spanish at igideasmath.com Find the value of. Round your answer to the nearest tenth WHT IF? In Eample, the length of the shorter leg is instead of. Show that the tangent of 60 is still equal to. Section 9. The Tangent Ratio

31 Solving Real-Life Problems The angle that an upward line of sight makes with a horizontal line is called the angle of elevation. Modeling with Mathematics You are measuring the height of a spruce tree. You stand feet from the base of the tree. You measure the angle of elevation from the ground to the top of the tree to be 9. Find the height h of the tree to the nearest foot. h ft 9 ft. Understand the Problem You are given the angle of elevation and the distance from the tree. You need to find the height of the tree to the nearest foot.. Make a Plan Write a trigonometric ratio for the tangent of the angle of elevation involving the height h. Then solve for h.. Solve the Problem tan 9 = opp. adj. Write ratio for tangent of 9. tan 9 = h Substitute. tan 9 = h Multiply each side by. 7.9 h Use a calculator. The tree is about 7 feet tall.. Look ack heck your answer. ecause 9 is close to 60, the value of h should be close to the length of the longer leg of a triangle, where the length of the shorter leg is feet. longer leg = shorter leg = Substitute Triangle Theorem 77.9 Use a calculator. The value of 77.9 feet is close to the value of h. h in in. Monitoring Progress Help in English and Spanish at igideasmath.com 6. You are measuring the height of a lamppost. You stand 0 inches from the base of the lamppost. You measure the angle of elevation from the ground to the top of the lamppost to be 70. Find the height h of the lamppost to the nearest inch. hapter 9 Right Triangles and Trigonometry

32 9. Eercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. OMPLETE THE SENTENE The tangent ratio compares the length of to the length of.. WRITING Eplain how you know the tangent ratio is constant for a given angle measure. Monitoring Progress and Modeling with Mathematics In Eercises 6, find the tangents of the acute angles in the right triangle. Write each answer as a fraction and as a decimal rounded to four decimal places. (See Eample.). R 8 T. G J H S. E 7 F 6. J L In Eercises 7 0, find the value of. Round your answer to the nearest tenth. (See Eample.) ERROR NLYSIS In Eercises and, describe the error in the statement of the tangent ratio. orrect the error if possible. Otherwise, write not possible. 6 K tan =.0 In Eercises and, use a special right triangle to find the tangent of the given angle measure. (See Eample.).. 0. MOELING WITH MTHEMTIS surveyor is standing 8 feet from the base of the Washington Monument. The surveyor measures the angle of elevation from the ground to the top of the monument to be 78. Find the height h of the Washington Monument to the nearest foot. (See Eample.) 6. MOELING WITH MTHEMTIS Scientists can measure the depths of craters on the moon by looking at photos of shadows. The length of the shadow cast by the edge of a crater is 00 meters. The angle of elevation of the rays of the Sun is. Estimate the depth d of the crater. Sun s ray. 7 E F tan = 7 00 m 7. USING STRUTURE Find the tangent of the smaller acute angle in a right triangle with side lengths,, and. d Section 9. The Tangent Ratio

33 8. USING STRUTURE Find the tangent of the larger acute angle in a right triangle with side lengths,, and. 9. RESONING How does the tangent of an acute angle in a right triangle change as the angle measure increases? Justify your answer. 0. RITIL THINKING For what angle measure(s) is the tangent of an acute angle in a right triangle equal to? greater than? less than? Justify your answer.. MKING N RGUMENT Your family room has a sliding-glass door. You want to buy an awning for the door that will be just long enough to keep the Sun out when it is at its highest point in the sky. The angle of elevation of the rays of the Sun at this point is 70, and the height of the door is 8 feet. Your sister claims you can determine how far the overhang should etend by multiplying 8 by tan 70. Is your sister correct? Eplain.. THOUGHT PROVOKING To create the diagram below, you begin with an isosceles right triangle with legs unit long. Then the hypotenuse of the first triangle becomes the leg of a second triangle, whose remaining leg is unit long. ontinue the diagram until you have constructed an angle whose tangent is. pproimate the measure of this angle. 6. PROLEM SOLVING Your class is having a class picture taken on the lawn. The photographer is positioned feet away from the center of the class. The photographer turns 0 to look at either end of the class. Sun s ray 8 ft 0 ft HOW O YOU SEE IT? Write epressions for the tangent of each acute angle in the right triangle. Eplain how the tangent of one acute angle is related to the tangent of the other acute angle. What kind of angle pair is and? a. RESONING Eplain why it is not possible to find the tangent of a right angle or an obtuse angle. c b Maintaining Mathematical Proficiency Find the value of. (Section 9.) a. What is the distance between the ends of the class? b. The photographer turns another 0 either way to see the end of the camera range. If each student needs feet of space, about how many more students can fit at the end of each row? Eplain. 6. PROLEM SOLVING Find the perimeter of the figure, where = 6, = F, and is the midpoint of. E 0 G Reviewing what you learned in previous grades and lessons 9. F H 6 hapter 9 Right Triangles and Trigonometry

34 9. The Sine and osine Ratios Essential Question How is a right triangle used to find the sine and cosine of an acute angle? Is there a unique right triangle that must be used? Let be a right triangle with acute. The sine of and cosine of (written as sin and cos, respectively) are defined as follows. length of leg opposite sin = length of hypotenuse = hypotenuse opposite cos = length of leg adjacent to = length of hypotenuse adjacent alculating Sine and osine Ratios Work with a partner. Use dynamic geometry software. a. onstruct, as shown. onstruct segments perpendicular to to form right triangles that share verte and are similar to with vertices, as shown. 6 0 K L M N O P Q J I H G F E Sample Points (0, 0) (8, 6) (8, 0) ngle m = 6.87 b. alculate each given ratio to complete the table for the decimal values of sin and cos for each right triangle. What can you conclude? Sine ratio K K LE L MF M NG N OH O PI P QJ Q sin osine ratio K E L F M G N H O I P J Q LOOKING FOR STRUTURE To be proficient in math, you need to look closely to discern a pattern or structure. cos ommunicate Your nswer. How is a right triangle used to find the sine and cosine of an acute angle? Is there a unique right triangle that must be used?. In Eploration, what is the relationship between and in terms of their measures? Find sin and cos. How are these two values related to sin and cos? Eplain why these relationships eist. Section 9. The Sine and osine Ratios 7

35 9. Lesson What You Will Learn ore Vocabulary sine, p. 8 cosine, p. 8 angle of depression, p. REING Remember the following abbreviations. sine sin cosine cos hypotenuse hyp. Use the sine and cosine ratios. Find the sine and cosine of angle measures in special right triangles. Solve real-life problems involving sine and cosine ratios. Use a trigonometric identity. Using the Sine and osine Ratios The sine and cosine ratios are trigonometric ratios for acute angles that involve the lengths of a leg and the hypotenuse of a right triangle. ore oncept Sine and osine Ratios Let be a right triangle with acute. The sine of and cosine of (written as sin and cos ) are defined as follows. length of leg opposite sin = length of hypotenuse cos = = length of leg adjacent to = length of hypotenuse leg opposite hypotenuse leg adjacent to Finding Sine and osine Ratios Find sin S, sin R, cos S, and cos R. Write each answer as a fraction and as a decimal rounded to four places. opp. S sin S = hyp. = RT adj. to S cos S = = ST hyp. SR = 6 6 SR = 6 opp. R sin R = = ST 6 hyp. SR = cos R = adj. to R hyp. R 6 6 = RT S 6 T SR = In Eample, notice that sin S = cos R and sin R = cos S. This is true because the side opposite S is adjacent to R and the side opposite R is adjacent to S. The relationship between the sine and cosine of S and R is true for all complementary angles. ore oncept Sine and osine of omplementary ngles The sine of an acute angle is equal to the cosine of its complement. The cosine of an acute angle is equal to the sine of its complement. Let and be complementary angles. Then the following statements are true. sin = cos(90 ) = cos sin = cos(90 ) = cos cos = sin(90 ) = sin cos = sin(90 ) = sin 8 hapter 9 Right Triangles and Trigonometry

36 Rewriting Trigonometric Epressions Write sin 6 in terms of cosine. Use the fact that the sine of an acute angle is equal to the cosine of its complement. sin 6 = cos(90 6 ) = cos The sine of 6 is the same as the cosine of. You can use the sine and cosine ratios to find unknown measures in right triangles. Finding Leg Lengths Find the values of and y using sine and cosine. Round your answers to the nearest tenth. 6 Step Use a sine ratio to find the value of. y sin 6 = opp. Write ratio for sine of 6. hyp. sin 6 = Substitute. sin 6 = Multiply each side by. 6. Use a calculator. The value of is about 6.. Step Use a cosine ratio to find the value of y. cos 6 = adj. hyp. Write ratio for cosine of 6. cos 6 = y Substitute. cos 6 = y Multiply each side by..6 y Use a calculator. The value of y is about.6. Monitoring Progress Help in English and Spanish at igideasmath.com. Find sin, sin F, cos, and cos F. Write each answer as a fraction and as a decimal rounded to four places. E 7 F. Write cos in terms of sine.. Find the values of u and t using sine and cosine. Round your answers to the nearest tenth. u 6 8 t Section 9. The Sine and osine Ratios 9

37 Finding Sine and osine in Special Right Triangles Find the sine and cosine of a angle. Finding the Sine and osine of egin by sketching a triangle. ecause all such triangles are similar, you can simplify your calculations by choosing as the length of each leg. Using the Triangle Theorem, the length of the hypotenuse is. STUY TIP Notice that sin = cos(90 ) = cos. sin = opp. hyp. cos = adj. hyp. = = = = Find the sine and cosine of a 0 angle. Finding the Sine and osine of 0 egin by sketching a triangle. ecause all such triangles are similar, you can simplify your calculations by choosing as the length of the shorter leg. Using the Triangle Theorem, the length of the longer leg is and the length of the hypotenuse is. 0 sin 0 = opp. hyp. cos 0 = adj. hyp. = = = Monitoring Progress Help in English and Spanish at igideasmath.com. Find the sine and cosine of a 60 angle. 0 hapter 9 Right Triangles and Trigonometry

38 Solving Real-Life Problems Recall from the previous lesson that the angle an upward line of sight makes with a horizontal line is called the angle of elevation. The angle that a downward line of sight makes with a horizontal line is called the angle of depression. Modeling with Mathematics You are skiing on a mountain with an altitude of 00 feet. The angle of depression is. Find the distance you ski down the mountain to the nearest foot. ft 00 ft Not drawn to scale. Understand the Problem You are given the angle of depression and the altitude of the mountain. You need to find the distance that you ski down the mountain.. Make a Plan Write a trigonometric ratio for the sine of the angle of depression involving the distance. Then solve for.. Solve the Problem sin = opp. hyp. sin = 00 Write ratio for sine of. Substitute. sin = 00 Multiply each side by. = 00 sin 8. ivide each side by sin. Use a calculator. You ski about 9 feet down the mountain.. Look ack heck your answer. The value of sin is about 0.8. Substitute for in the sine ratio and compare the values This value is approimately the same as the value of sin. Monitoring Progress Help in English and Spanish at igideasmath.com. WHT IF? In Eample 6, the angle of depression is 8. Find the distance you ski down the mountain to the nearest foot. Section 9. The Sine and osine Ratios

39 STUY TIP Note that sin θ represents (sin θ) and cos θ represents (cos θ). Using a Trigonometric Identity In the figure, the point (, y) is on a circle of radius with center at the origin. onsider an angle θ (the Greek letter theta) with its verte at the origin. You know that = cos θ and y = sin θ. y the Pythagorean Theorem, So, + y =. cos θ + sin θ =. y θ (, y) y The equation cos θ + sin θ = is true for any value of θ. In this section, θ will always be an acute angle. trigonometric equation that is true for all values of the variable for which both sides of the equation are defined is called a trigonometric identity. ecause of its derivation above, cos θ + sin θ = is also referred to as a Pythagorean identity. Finding Trigonometric Values Given that cos θ =, find sin θ and tan θ. Step Find sin θ. sin θ + cos θ = Write Pythagorean identity. sin θ + ( ) = Substitute for cos θ. sin θ + 9 = Evaluate power. sin θ = 9 sin θ = 6 Subtract 9 from each side. Subtract. STUY TIP Using the diagram above, you can write tan θ = y = sin θ cos θ. sin θ = Step Find tan θ using sin θ and cos θ. tan θ = sin θ cos θ = Take the positive square root of each side. = Monitoring Progress Help in English and Spanish at igideasmath.com 6. Given that sin θ =, find cos θ and tan θ. hapter 9 Right Triangles and Trigonometry

40 9. Eercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. VOULRY The sine ratio compares the length of to the length of.. VOULRY The ratio compares the length of the adjacent leg to the length of the hypotenuse.. WRITING How are angles of elevation and angles of depression similar? How are they different?. WHIH ONE OESN T ELONG? Which ratio does not belong with the other three? Eplain your reasoning. Monitoring Progress and Modeling with Mathematics sin cos tan In Eercises 0, find sin, sin E, cos, and cos E. Write each answer as a fraction and as a decimal rounded to four places. (See Eample.) F E E 6. F E F 0. In Eercises, write the epression in terms of cosine. (See Eample.). sin 7. sin 8. sin 9. sin 6 E E F 7 F F 8 E In Eercises 8, write the epression in terms of sine.. cos 9 6. cos 7. cos 7 8. cos 8 In Eercises 9, find the value of each variable using sine and cosine. Round your answers to the nearest tenth. (See Eample.) 9... w b 8 y a v 0... Section 9. The Sine and osine Ratios 6 r m 0 p n s 8 6 q

41 . RESONING Which ratios are equal? Select all that apply. (See Eample.) X Y Z 0. ERROR NLYSIS escribe and correct the error in finding sin. sin = sin X sin Z cos X cos Z. MOELING WITH MTHEMTIS The top of the slide is feet from the ground and has an angle of depression of. What is the length of the slide? (See Eample 6.) 6. RESONING Which ratios are equal to? Select all that apply. (See Eample.) J ft K 0 L. MOELING WITH MTHEMTIS Find the horizontal distance the escalator covers. sin L sin J cos L cos J ft 6 ft 7. RESONING Write the trigonometric ratios that have a value of. N M 0 60 P 8. RESONING Write the trigonometric ratios that have a value of. 8 Q R S In Eercises 8, find sin θ and tan θ. (See Eample 7.). cos θ =. cos θ = 9 7. cos θ = 7 8. cos θ = 7 6. cos θ = 8 8. cos θ = 9 0 In Eercises 9, find cos θ and tan θ. 9. sin θ = 0. sin θ = 9. WRITING Eplain how to tell which side of a right triangle is adjacent to an angle and which side is the hypotenuse.. sin θ = 8. sin θ = 9. sin θ = 7 0. sin θ = 6 7 hapter 9 Right Triangles and Trigonometry

42 . ERROR NLYSIS escribe and correct the error in finding cos θ when sin θ = ERROR NLYSIS escribe and correct the error in finding sin θ when cos θ = 8. cos θ sin θ = cos θ ( 8 9 ) = cos θ 6 8 = 7. PROLEM SOLVING You are flying a kite with 0 feet of string etended. The angle of elevation from the spool of string to the kite is 67. a. raw and label a diagram that represents the situation. b. How far off the ground is the kite if you hold the spool feet off the ground? escribe how the height where you hold the spool affects the height of the kite. 8. MKING N RGUMENT Your friend uses the equation sin 9 = to find. Your cousin uses 6 the equation cos = to find. Who is correct? 6 Eplain your reasoning. cos θ = cos θ = 8 cos θ = 9 sin θ + cos θ = sin θ + 8 = sin θ = 8 sin θ = 7 sin θ = y 9. MOELING WITH MTHEMTIS Planes that fly at high speeds and low elevations have radar systems that can determine the range of an obstacle and the angle of elevation to the top of the obstacle. The radar of a plane flying at an altitude of 0,000 feet detects a tower that is,000 feet away, with an angle of elevation of.,000 ft h ft Not drawn to scale a. How many feet must the plane rise to pass over the tower? b. Planes cannot come closer than 000 feet vertically to any object. t what altitude must the plane fly in order to pass over the tower? 0. WRITING escribe what you must know about a triangle in order to use the sine ratio and what you must know about a triangle in order to use the cosine ratio.. MTHEMTIL ONNETIONS If EQU is equilateral and RGT is a right triangle with RG =, RT =, and m T = 90, show that sin E = cos G.. MOELING WITH MTHEMTIS Submarines use sonar systems, which are similar to radar systems, to detect obstacles. Sonar systems use sound to detect objects under water. 000 m Not drawn to scale 9 00 m a. You are traveling underwater in a submarine. The sonar system detects an iceberg 000 meters ahead, with an angle of depression of to the bottom of the iceberg. How many meters must the submarine lower to pass under the iceberg? b. The sonar system then detects a sunken ship 00 meters ahead, with an angle of elevation of 9 to the highest part of the sunken ship. How many meters must the submarine rise to pass over the sunken ship? Section 9. The Sine and osine Ratios

43 . RESONING Use sin = opp. hyp. show that tan = sin cos. and cos = adj. hyp. to. HOW O YOU SEE IT? Using only the given information, would you use a sine ratio or a cosine ratio to find the length of the hypotenuse? Eplain your reasoning. 6. THOUGHT PROVOKING One of the following infinite series represents sin and the other one represents cos (where is measured in radians). Which is which? Justify your answer. Then use each series to approimate the sine and cosine of π 6. (Hints: π = 80 ;! = ; Find the values that the sine and cosine ratios approach as the angle measure approaches zero.) 9 9 a.! +! 7 7! +... b.! +! 6 6! MULTIPLE REPRESENTTIONS You are standing on a cliff above an ocean. You see a sailboat from your vantage point 0 feet above the ocean. a. raw and label a diagram of the situation. b. Make a table showing the angle of depression and the length of your line of sight. Use the angles 0, 0, 60, 70, and 80. c. Graph the values you found in part (b), with the angle measures on the -ais. d. Predict the length of the line of sight when the angle of depression is STRT RESONING Make a conjecture about how you could use trigonometric ratios to find angle measures in a triangle. 8. RITIL THINKING Eplain why the area of in the diagram can be found using the formula rea = ab sin. Then calculate the area when a =, b = 7, and m = 0. a h b c Maintaining Mathematical Proficiency Find the value of. Tell whether the side lengths form a Pythagorean triple. (Section 9.) Use the two-way table to create a two-way table that shows the joint and marginal relative frequencies. (Section.) Gender Male Female Total Sophomore 6 80 Reviewing what you learned in previous grades and lessons 9 lass Junior 8 80 Senior 0 0 Total hapter 9 Right Triangles and Trigonometry

44 9.6 Solving Right Triangles Essential Question When you know the lengths of the sides of a right triangle, how can you find the measures of the two acute angles? Solving Special Right Triangles Work with a partner. Use the figures to find the values of the sine and cosine of and. Use these values to find the measures of and. Use dynamic geometry software to verify your answers. a. b. TTENING TO PREISION To be proficient in math, you need to calculate accurately and efficiently, epressing numerical answers with a degree of precision appropriate for the problem contet. 0 0 Solving Right Triangles Work with a partner. You can use a calculator to find the measure of an angle when you know the value of the sine, cosine, or tangent of the angle. Use the inverse sine, inverse cosine, or inverse tangent feature of your calculator to approimate the measures of and to the nearest tenth of a degree. Then use dynamic geometry software to verify your answers. a. b ommunicate Your nswer. When you know the lengths of the sides of a right triangle, how can you find the measures of the two acute angles?. ladder leaning against a building forms a right triangle with the building and the ground. The legs of the right triangle (in meters) form a -- Pythagorean triple. Find the measures of the two acute angles to the nearest tenth of a degree. Section 9.6 Solving Right Triangles 7

45 9.6 Lesson What You Will Learn ore Vocabulary inverse tangent, p. 8 inverse sine, p. 8 inverse cosine, p. 8 solve a right triangle, p. 9 Use inverse trigonometric ratios. Solve right triangles. Using Inverse Trigonometric Ratios Identifying ngles from Trigonometric Ratios etermine which of the two acute angles has a cosine of 0.. Find the cosine of each acute angle. adj. to cos = = hyp. adj. to cos = = hyp. = 0. The acute angle that has a cosine of 0. is. If the measure of an acute angle is 60, then its cosine is 0.. The converse is also true. If the cosine of an acute angle is 0., then the measure of the angle is 60. So, in Eample, the measure of must be 60 because its cosine is 0.. REING The epression tan is read as the inverse tangent of. NOTHER WY You can use the Table of Trigonometric Ratios available at igideasmath.com to approimate tan 0.7 to the nearest degree. Find the number closest to 0.7 in the tangent column and read the angle measure at the left. ore oncept Inverse Trigonometric Ratios Let be an acute angle. Inverse Tangent If tan =, then tan = m. Inverse Sine If sin = y, then sin y = m. Finding ngle Measures Let,, and be acute angles. Use a calculator to approimate the measures of,, and to the nearest tenth of a degree. a. tan = 0.7 b. sin = 0.87 c. cos = 0. a. m = tan b. m = sin c. m = cos tan sin = m = m Inverse osine If cos = z, then cos z = m. cos = m E Monitoring Progress Help in English and Spanish at igideasmath.com etermine which of the two acute angles has the given trigonometric ratio. F. The sine of the angle is.. The tangent of the angle is. 8 hapter 9 Right Triangles and Trigonometry

46 Monitoring Progress Solving Right Triangles ore oncept Help in English and Spanish at igideasmath.com Let G, H, and K be acute angles. Use a calculator to approimate the measures of G, H, and K to the nearest tenth of a degree.. tan G = 0.. sin H = cos K = 0.9 Solving a Right Triangle To solve a right triangle means to find all unknown side lengths and angle measures. You can solve a right triangle when you know either of the following. two side lengths one side length and the measure of one acute angle Solving a Right Triangle Solve the right triangle. Round decimal answers to the nearest tenth. NOTHER WY You could also have found m first by finding tan 6.. Step Use the Pythagorean Theorem to find the length of the hypotenuse. c c = a + b Pythagorean Theorem c = + Substitute. c = Simplify. c = Find the positive square root. c.6 Use a calculator. Step Find m. m = tan.7 Use a calculator. Step Find m. ecause and are complements, you can write m = 90 m 90.7 = 6.. In, c.6, m.7, and m 6.. Monitoring Progress Help in English and Spanish at igideasmath.com Solve the right triangle. Round decimal answers to the nearest tenth G 0 09 F E H 9 J Section 9.6 Solving Right Triangles 9

47 Solving a Right Triangle Solve the right triangle. Round decimal answers to the nearest tenth. H g Use trigonometric ratios to find the values of g and h. sin H = opp. hyp. J h G cos H = adj. hyp. REING raked stage slants upward from front to back to give the audience a better view. sin = h cos = g sin = h cos = g. h.8 g ecause H and G are complements, you can write m G = 90 m H = 90 = 6. In GHJ, h., g.8, and m G = 6. Solving a Real-Life Problem Your school is building a raked stage. The stage will be 0 feet long from front to back, with a total rise of feet. You want the rake (angle of elevation) to be or less for safety. Is the raked stage within your desired range? 0 ft stage back ft stage front Use the inverse sine ratio to find the degree measure of the rake. sin 0.8 The rake is about.8, so it is within your desired range of or less. Monitoring Progress 8. Solve the right triangle. Round decimal answers to the nearest tenth. 9. WHT IF? In Eample, suppose another raked stage is 0 feet long from front to back with a total rise of feet. Is the raked stage within your desired range? Help in English and Spanish at igideasmath.com Y X 8. Z 60 hapter 9 Right Triangles and Trigonometry

48 9.6 Eercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. OMPLETE THE SENTENE To solve a right triangle means to find the measures of all its and.. WRITING Eplain when you can use a trigonometric ratio to find a side length of a right triangle and when you can use the Pythagorean Theorem. Monitoring Progress and Modeling with Mathematics In Eercises 6, determine which of the two acute angles has the given trigonometric ratio. (See Eample.). The cosine of the. The sine of the angle is. angle is The sine of the 6. The tangent of the angle is angle is In Eercises 7, let be an acute angle. Use a calculator to approimate the measure of to the nearest tenth of a degree. (See Eample.) 7. sin = sin = cos = cos = 0.6. tan = 0.8. tan = 0.7 In Eercises 8, solve the right triangle. Round decimal answers to the nearest tenth. (See Eamples and.).. E. Y Z 6. G H 7. X M 8 0 K L 8. R S 7 9. ERROR NLYSIS escribe and correct the error in using an inverse trigonometric ratio. T 9 7 V 8 U 6 T sin 8 = m T 0. PROLEM SOLVING In order to unload clay easily, the body of a dump truck must be elevated to at least. The body of a dump truck that is feet long has been raised 8 feet. Will the clay pour out easily? Eplain your reasoning. (See Eample.). PROLEM SOLVING You are standing on a footbridge that is feet above a lake. You look down and see a duck in the water. The duck is 7 feet away from the footbridge. What is the angle of elevation from the duck to you? ft J 9 8 F 7 ft Section 9.6 Solving Right Triangles 6

49 . HOW O YOU SEE IT? Write three epressions that can be used to approimate the measure of. Which epression would you choose? Eplain your choice.. MOELING WITH MTHEMTIS The Uniform Federal ccessibility Standards specify that a wheelchair ramp may not have an incline greater than.76. You want to build a ramp with a vertical rise of 8 inches. You want to minimize the horizontal distance taken up by the ramp. raw a diagram showing the approimate dimensions of your ramp.. MOELING WITH MTHEMTIS The horizontal part of a step is called the tread. The vertical part is called the riser. The recommended riser-to-tread ratio is 7 inches : inches. a. Find the value of for stairs built using the recommended riser-to-tread ratio. tread riser b. You want to build stairs that are less steep than the stairs in part (a). Give an eample of a riser-totread ratio that you could use. Find the value of for your stairs.. USING TOOLS Find the measure of R without using a protractor. Justify your technique. Q 6. MKING N RGUMENT Your friend claims that tan =. Is your friend correct? Eplain your tan reasoning. USING STRUTURE In Eercises 7 and 8, solve each triangle. 7. JKM and LKM J 9 ft K M 8. TUS and VTW T U m m 6 S ft V 9 m W 9. MTHEMTIL ONNETIONS Write an epression that can be used to find the measure of the acute angle formed by each line and the -ais. Then approimate the angle measure to the nearest tenth of a degree. a. y = b. y = + 0. THOUGHT PROVOKING Simplify each epression. Justify your answer. a. sin (sin ) b. tan(tan y) c. cos(cos z). RESONING Eplain why the epression sin (.) does not make sense. L P Maintaining Mathematical Proficiency Solve the equation. (Skills Review Handbook). =. 9 = 8 R.. USING STRUTURE The perimeter of rectangle is 6 centimeters, and the ratio of its width to its length is :. Segment divides the rectangle into two congruent triangles. Find the side lengths and angle measures of these two triangles. Reviewing what you learned in previous grades and lessons. = =.9 6 hapter 9 Right Triangles and Trigonometry

50 What id You Learn? ore Vocabulary trigonometric ratio, p. cosine, p. 8 inverse cosine, p. 8 tangent, p. angle of depression, p. solve a right triangle, p. 9 angle of elevation, p. inverse tangent, p. 8 sine, p. 8 inverse sine, p. 8 ore oncepts Section 9. Tangent Ratio, p. Section 9. Sine and osine Ratios, p. 8 Sine and osine of omplementary ngles, p. 8 Using a Trigonometric Identity, p. Section 9.6 Inverse Trigonometric Ratios, p. 8 Solving a Right Triangle, p. 9 Mathematical Practices. In Eercise on page 6, your brother claims that you could determine how far the overhang should etend by dividing 8 by tan 70. Justify his conclusion and eplain why it works.. In Eercise 7 on page, eplain the flaw in the argument that the kite is 8. feet high. Performance Task: hallenging the Rock Wall There are multiple ways to use indirect measurement when you cannot measure directly. When you look at a rock wall, do you wonder about its height before you attempt to climb it? What other situations require you to calculate a height using trigonometry and indirect measurement? To eplore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at igideasmath.com. 6

51 9 hapter Review 9. The Pythagorean Theorem (pp. 7 ) ynamic Solutions available at igideasmath.com Find the value of. Then tell whether the side lengths form a Pythagorean triple. 0 c = a + b = + 0 = + 00 = 6 = Pythagorean Theorem Substitute. Multiply. dd. Find the positive square root. The value of is. ecause the side lengths, 0, and are integers that satisfy the equation c = a + b, they form a Pythagorean triple. Find the value of. Then tell whether the side lengths form a Pythagorean triple Verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse?. 6, 8, and 9. 0,, and 6 6., 8, and 9. Special Right Triangles (pp. 0) Find the value of. Write your answer in simplest form. y the Triangle Sum Theorem, the measure of the third angle must be, so the triangle is a triangle. hypotenuse = leg Triangle Theorem = 0 Substitute. = 0 Simplify. 0 The value of is 0. Find the value of. Write your answer in simplest form hapter 9 Right Triangles and Trigonometry

52 9. Similar Right Triangles (pp. 8) Identify the similar triangles. Then find the value of. Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. + y the Geometric Mean (ltitude) Theorem, you know that is the geometric mean of and. = Geometric Mean (ltitude) Theorem 6 = Square. 8 = ivide each side by. The value of is 8. Identify the similar triangles. Then find the value of. 0. F. J K. T E 6 H G M L S 6 V U 9. The Tangent Ratio (pp. 6) Find tan M and tan N. Write each answer as a fraction and as a decimal rounded to four places. N opp. M tan M = adj. to M = LN LM = 6 8 = = tan N = opp. N adj. to N = LM LN = 8 6 =. M 8 L. Find tan J and tan L. Write each answer as a fraction and as a decimal rounded to four decimal places.. The angle between the bottom of a fence and the top of a tree is 7. The tree is feet from the fence. How tall is the tree? Round your answer to the nearest foot. J 6 L 60 K 7 ft hapter 9 hapter Review 6

53 9. The Sine and osine Ratios (pp. 7 6) Find sin, sin, cos, and cos. Write each answer as a fraction and as a decimal rounded to four places. opp. sin = = hyp. = 0 = opp. sin = = hyp. = 6 = cos = adj. to = hyp. = 6 = 8 7 adj. to cos = = hyp. 0 = 0 = Find sin X, sin Z, cos X, and cos Z. Write each answer as a fraction and as a decimal rounded to four decimal places. X 0 Y 7 9 Find the value of each variable using sine and cosine. Round your answers to the nearest tenth. Z v t s r w 70 s Write sin 7 in terms of cosine. 0. Given that sin θ =, find cos θ and tan θ Solving Right Triangles (pp. 7 6) Solve the right triangle. Round decimal answers to the nearest tenth. Step Use the Pythagorean Theorem to find the length of the hypotenuse. c = a + b Pythagorean Theorem c = 9 + Substitute. c = 0 Simplify. c = 0 Find the positive square root. c. Use a calculator. c 9 Step Step Find m. m = tan. Use a calculator. 9 Find m. ecause and are complements, you can write m = 90 m 90. = 7.7. In, c., m., and m 7.7. Solve the right triangle. Round decimal answers to the nearest tenth... N 6 M. 7 Z 8 0 L X Y 66 hapter 9 Right Triangles and Trigonometry

54 9 hapter Test Find the value of each variable. Round your answers to the nearest tenth.. t 8 s. y 6. j 0 0 k Verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse?. 6, 0, and., 67, and 9 6.,, and. Solve. Round decimal answers to the nearest tenth Find the geometric mean of 9 and.. Write cos in terms of sine. Find the value of each variable. Write your answers in simplest form.. 6 q r. c 6 e d. h 0 8 f. Given that cos θ =, find sin θ and tan θ You are given the measures of both acute angles of a right triangle. an you determine the side lengths? Eplain. 7. You are at a parade looking up at a large balloon floating directly above the street. You are 60 feet from a point on the street directly beneath the balloon. To see the top of the balloon, you look up at an angle of. To see the bottom of the balloon, you look up at an angle of 9. Estimate the height h of the balloon. h viewing angle 9 60 ft 8. You want to take a picture of a statue on Easter Island, called a moai. The moai is about feet tall. Your camera is on a tripod that is feet tall. The vertical viewing angle of your camera is set at 90. How far from the moai should you stand so that the entire height of the moai is perfectly framed in the photo? hapter 9 hapter Test 67

55 9 umulative ssessment. The size of a laptop screen is measured by the length of its diagonal. You want to purchase a laptop with the largest screen possible. Which laptop should you buy? in. 0 in. 9 in.. in. in. 8 in. 6.7 in. 6 in.. In PQR and SQT, S is between P and Q, T is between R and Q, and QS SP = QT TR. What must be true about ST and PR? Select all that apply. ST PR ST PR ST = PR ST = PR. In the diagram, JKL QRS. hoose the symbol that makes each statement true. J Q 0 6 R K 8 L S sin J sin Q sin L cos J cos L tan Q cos S cos J cos J sin S tan J tan Q tan L tan Q tan S cos Q sin Q cos L < = >. Out of 60 students, 6 like fiction novels, 0 like nonfiction novels, and like both. What is the probability that a randomly selected student likes fiction or nonfiction novels? 68 hapter 9 Right Triangles and Trigonometry

56 . reate as many true equations as possible. X 60 6 Y = Z sin X cos X tan X XY YZ XZ XZ sin Z cos Z tan Z XY YZ YZ XY 6. Prove that quadrilateral EFG is a kite. Given HE HG, EG F Prove FE FG, E G H E F 7. The area of the triangle is square inches. Find the value of. in. G ( + 8) in. 8. lassify the solution(s) of each equation as real numbers, imaginary numbers, or pure imaginary numbers. Justify your answers. a. + 6 = 0 b. ( i ) ( i + 6) = 8 + c. = 0 d. + = e. = 6 f. 8 = 0 9. The Red Pyramid in Egypt has a square base. Each side of the base measures 7 feet. The height of the pyramid is feet. a. Use the side length of the base, the height of the pyramid, and the Pythagorean Theorem to find the slant height,, of the pyramid. b. Find. c. Name three possible ways of finding m. Then, find m. hapter 9 umulative ssessment 69