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1 Algebra Lab Investigating Trigonometri Ratios You an use paper triangles to investigate the ratios of the lengths of sides of right triangles. Virginia SOL Preparation for G.8 The student will solve realworld problems involving right triangles by using the Pythagorean Theorem and its onverse, properties of speial right triangles, and right triangle trigonometry. Ativity Collet the Data Step 1 Use a ruler and grid paper to draw several right triangles with legs in a ratio of :8. Inlude right triangles with the side lengths listed in the table below and several more right triangles similar to these three. Label the verties of eah triangle as A, B, and C, where C is at the right angle, B is opposite the longest leg, and A is opposite the shortest leg. A 16 Step 2 Copy the table below. Complete the first three olumns by measuring the hypotenuse (side AB ) in eah right triangle you reated and reording its length to the nearest tenth. C 10 B Step Calulate and reord the ratios in the middle two olumns. Round to the nearest hundredth. Step 4 Use a protrator to arefully measure angles A and B to the nearest degree in eah right triangle. Reord the angle measures in the table. Side Lengths Ratios Angle Measures side BC side AC side AB _ BC _ BC angle A angle B angle C AC AB Analyze the Results 1. Examine the measures and ratios in the table. What do you notie? Write a sentene or two to desribe any patterns you see. Make a Conjeture 2. For any right triangle similar to the ones you have drawn here, what will be the value of the ratio of the length of the shortest leg to the length of the longest leg?. If you draw a right triangle and alulate the ratio of the length of the shortest leg to the length of the hypotenuse to be approximately 0., what will be the measure of the larger aute angle in the right triangle? x=? 648 Explore 10-8 Algebra Lab: Investigating Trigonometri Ratios

2 Trigonometri Ratios Then You used the Pythagorean Theorem (Lesson 10-) Now 1Find trigonometri ratios of angles. 2Use trigonometry to solve triangles. Why? If a road has a perent grade of 8%, this means the road rises or falls 8 feet over a horizontal distane of 100 feet. Trigonometri ratios an be used to determine the angle that the road rises or falls. New Voabulary trigonometry trigonometri ratio sine osine tangent solving the triangle inverse sine inverse osine inverse tangent Trigonometri Ratios Trigonometry is the study of relationships among the 1 angles and sides of triangles. A trigonometri ratio is a ratio that ompares the side lengths of two sides of a right triangle. The three most ommon trigonometri ratios, sine, osine, and tangent, are desribed below. Key Conept Trigonometri Ratios leg opposite A sine of A = hypotenuse Words Symbols Model sin A = a _ A leg adjaent to A osine of A = hypotenuse os A = b _ b Virginia i SOL Preparation for G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its onverse, properties of speial right triangles, and right triangle trigonometry. leg opposite A tangent of A = Leg adjaent to A tan A = a _ b Opposite, adjaent, and hypotenuse are abbreviated opp, adj, and hyp, respetively Example 1 Find Sine, Cosine, and Tangent Ratios Find the values of the three trigonometri ratios for angle A. Step 1 Use the Pythagorean Theorem to find b. a 2 + b 2 = 2 Pythagorean Theorem Step b 2 = 1 2 a = 9 and = b 2 = 22 Simplify. b 2 = 144 b = 12 Subtrat 81 from eah side. Take the square root of eah side. Use the side lengths to write the trigonometri ratios. sin A = _ opp hyp = _ 9 1 = _ os A = _ adj hyp = _ 12 1 = 4_ tan A = _ opp adj 9 C b a 1 B = _ 9 12 = _ 4 GuidedPratie 1. Find the values of the three trigonometri ratios for angle B. onneted.mgraw-hill.om 649

3 Wath Out! Calulator Mode Make sure your graphing alulator is in degree mode. Example 2 Use a Calulator to Evaluate Expressions Use a alulator to find os 42 to the nearest ten-thousandth. KEYSTROKES: 42 Rounded to the nearest ten-thousandth, os GuidedPratie 2A. sin 1 2B. tan 6 2C. os Use Trigonometri Ratios When you find all unknown measures of the sides 2 and angles of a right triangle, you are solving the triangle. You an find the missing measures if you know the measure of two sides of the triangle or the measure of one side and the measure of one aute angle. Example Solve a Triangle Solve the right triangle. Round eah side length to the nearest tenth. 6 Study Tip Remembering Trigonometri Ratios SOH CAH TOA an be used to help you remember the ratios for sine, osine, and tangent. Eah letter represents a word. sin A = _ opp hyp os A = _ adj hyp tan A = _ opp adj Step 1 Find the measure of A ( ) = 49 The measure of A = 49. Step 2 Find a. Sine you are given the measure of the side opposite B and are finding the measure of the side adjaent to B, use the tangent ratio. tan 41 = 6 _ a Definition of tangent a tan 41 = 6 Multiply eah side by a. a = 6_ or about 6.9 tan 41 So the measure of a or BC is about 6.9. a 41 Divide eah side by tan 41. Use a alulator. Step Find. Sine you are given the measure of the side opposite B and are finding the measure of the hypotenuse, use the sine ratio. sin 41 = 6 _ Definition of sine sin 41 = 6 Multiply eah side by. = 6_ or about 9.1 sin 41 So the measure of or AB is about 9.1. Divide eah side by sin 41. Use a alulator. GuidedPratie A. B Lesson 10-8 Trigonometri Ratios

4 Real-World Example 4 Find a Missing Side Length EXERCISE A trainer sets the inline on a treadmill to 10. The walking surfae of the treadmill is feet long. About how many inhes is the end of the treadmill from the floor? sin 10 = _ h ft h Definition of sine sin 10 = h Multiply eah side by. 0.8 h Use a alulator. The value of h is in feet. Multiply 0.8 by 12 to onvert feet to inhes. The trainer raised the treadmill about 10.4 inhes. 10 Real-World Link For optimum health, all adults ages 18 6 should get at least 0 minutes of moderately intense ativity five days per week. Soure: Amerian Heart Assoiation GuidedPratie 4. SKATEBOARDING The angle that a skateboarding ramp forms with the ground is 2 and the height of the ramp is 6 feet. Determine the length of the ramp. A trigonometri funtion has a rule given by a trigonometri ratio. If you know the sine, osine, or tangent of an aute angle, you an use the inverse of the trigonometri funtion to find the measure of the angle. Key Conept Inverse Trigonometri Funtions Words If A is an aute angle and the sine of A is x, then the inverse sine of x is the measure of A. Symbols Words Symbols Words Symbols If sin A = x, then sin -1 x = m A. If A is an aute angle and the osine of A is x, then the inverse osine of x is the measure of A. If os A = x, then os -1 x = m A. If A is an aute angle and the tangent of A is x, then the inverse tangent of x is the measure of A. If tan A = x, then tan -1 x = m A. Example Find a Missing Angle Measure Find m Y to the nearest degree. You know the measure of the side adjaent to Y and 19 the measure of the hypotenuse. Use the osine ratio. 8 os Y = _ 8 Definition of osine 19 Use a alulator and the [CO S -1 ] funtion to find the measure of the angle. KEYSTROKES: [CO S -1 ] So, m Y = 6. GuidedPratie. Find m X to the nearest degree if XY = 14 and YZ =. onneted.mgraw-hill.om 61

5 Chek Your Understanding = Step-by-Step Solutions begin on page R12. Example 1 Find the values of the three trigonometri ratios for angle A Example 2 Example Use a alulator to find the value of eah trigonometri ratio to the nearest ten-thousandth.. sin 6. os 2. tan os 82 Solve eah right triangle. Round eah side length to the nearest tenth Example 4 1. SNOWBOARDING A hill used for snowboarding has a vertial drop of 00 feet. The angle the run makes with the ground is 18. Estimate the length of r. 00 ft r 18 Example Find m X for eah right triangle to the nearest degree Lesson 10-8 Trigonometri Ratios

6 Pratie and Problem Solving Extra Pratie begins on page 81. Example 1 Find the values of the three trigonometri ratios for angle B Example 2 Use a alulator to find the value of eah trigonometri ratio to the nearest ten-thousandth. 21. tan sin os tan 4 Example 2. sin 26. os sin tan 60 Solve eah right triangle. Round eah side length to the nearest tenth Example 4. ESCALATORS At a loal mall, an esalator is 110 feet long. The angle the esalator makes with the ground is 29. Find the height reahed by the esalator. 110 ft h 29 Example Find m J for eah right triangle to the nearest degree B 42. MONUMENTS The Linoln Memorial building measures 204 feet long, 14 feet wide, and 99 feet tall. Chloe is looking at the top of the monument at an angle of. How far away is she standing from the monument? onneted.mgraw-hill.om 6

7 4 AIRPLANES Ella looks down at a ity from an airplane window. The airplane is 000 feet in the air, and she looks down at an angle of 8. Determine the horizontal distane to the ity. 44. FORESTS A forest ranger estimates the height of a tree is about 1 feet. If the forest ranger is standing 100 feet from the base of the tree, what is the measure of the angle formed by the ranger and the top of the tree? Suppose A is an aute angle of right triangle ABC. 4. Find sin A and tan A if os A = _ Find tan A and os A if sin A = 2_. 4. Find os A and tan A if sin A = 1_ Find sin A and os A if tan A = _. 49. SUBMARINES A submarine desends into the oean at an angle of 10 below the water line and travels miles diagonally. How far beneath the surfae of the water has the submarine reahed? C 0. MULTIPLE REPRESENTATIONS In this problem, you will explore a relationship between the sine and osine funtions a. Tabular Copy and omplete the table using the triangles shown above Triangle Trigonometri Ratios sin 2 os 2 sin 2 + os 2 = ABC JKL XYZ sin A = os A = sin 2 A = os 2 A = sin C = os C = sin 2 C = os 2 C = sin J = os J = sin 2 J = os 2 J = sin L = os L = sin 2 L = os 2 L = sin X = os X = sin 2 X = os 2 X = sin Z = os Z = sin 2 Z = os 2 Z = b. Verbal Make a onjeture about the sum of the squares of the sine and osine funtions of an aute angle in a right triangle. H.O.T. Problems Use Higher-Order Thinking Skills 1. CHALLENGE Find a and in the triangle shown. 2. REASONING Use the definitions of the sine and osine ratios to define the tangent ratio. 64 Lesson 10-8 Trigonometri Ratios + (6a - ) (12a + ) - 2. OPEN ENDED Write a problem that uses the osine ratio to find the measure of an unknown angle in a triangle. Then solve the problem. 4. REASONING The sine and osine of an aute angle in a right triangle are equal. What an you onlude about the triangle?. WRITING IN MATH Explain how to use trigonometri ratios to find the missing length of a side of a right triangle given the measure of one aute angle and the length of one side.

8 Virginia SOL Pratie A.., G.8 6. Whih graph below represents the solution set for -2 x 4? A B C D PROBABILITY Suppose one hip is hosen from a bin with the hips shown. To the nearest tenth, what is the probability that a green hip is hosen? Color Number yellow blue 9 orange green red 6 F 0.2 H 0.6 G 0. J In the graph, for what value(s) of x is y = 0? A 0 C 1 B -1 D 1 and EXTENDED RESPONSE A 16-foot ladder is plaed against the side of a house so that the bottom of the ladder is 8 feet from the base of the house. a. If the bottom of the ladder is moved loser to the base of the house, does the height reahed by the ladder inrease or derease? b. What onlusion an you make about the distane between the bottom of the ladder and the base of the house and the height reahed by the ladder?. How high does the ladder reah if the ladder is feet from the base of the house? y x Spiral Review For eah set of measures given, find the measures of the missing sides if ABC DFH. (Lesson 10-) 60. a = 16, b = 12, = 8, f = 6 a h d 61. d = 9, f = 6, h = 4, b = a = 6, b = 21, h = 11, f = 14 b f 6. = 22., b = 20, h = 9, d = 2 Find the oordinates of the midpoint of the segment with the given endpoints. (Lesson 10-6) 64. (, ), (11, 9) 6. (8, 2), (6, 4) 66. (-1, ), (1, -) 6. FINANCIAL LITERACY A salesperson is paid $2,000 a year plus % of the amount in sales made. What is the amount of sales needed to have an annual inome greater than $4,000? (Lesson -) Skills Review Solve eah proportion. (Lesson 2-6) 68. _ 8 9 = _ 6 z 69. _ p 6 = 4_ 0. _ 0. r = _ _ = y_ 8.4 onneted.mgraw-hill.om 6