UNIT 10 Trigonometry UNIT OBJECTIVES 287
|
|
- Adele Simpson
- 6 years ago
- Views:
Transcription
1 UNIT 10 Trigonometry Literally translated, the word trigonometry means triangle measurement. Right triangle trigonometry is the study of the relationships etween the side lengths and angle measures of right triangles. Trigonometry is one of the most useful sujects in all of geometry ecause of its many real-world applications. stronomy, medical imaging, meteorology, cartography, and computer graphics are just a few of the fields in which it is used. Trigonometry is also very useful in finding indirect measurements. During The Great Trigonometric Survey of the 1800s, trigonometry was used to find land measurements necessary for mapmaking. Indirect measurement was used to find the heights of these Himalayan mountains: K2, Kanchenjunga, and Mount Everest. In this unit, you will study the three asic trigonometric ratios sine, cosine, and tangent and use them to solve prolems involving oth right triangles and all other triangles. UNIT OJETIVES Define the sine, cosine, and tangent ratios. Use the sine, cosine, and tangent ratios to find missing angle measures and missing side lengths in right triangles. Identify trigonometric identities. Use the relationships in and triangles to find trigonometric ratios and use those ratios to solve prolems. Use the Laws of Sines and osines to find missing angle or side measures in triangles. unit 10 TRIGONOMETRY 287
2
3 Tangents Ojectives Define the tangent ratio and epress tangent ratios as fractions or decimals. Use a calculator to find the value of the tangent of an angle. Sometimes guideooks give the grade of a hiking trail. What does it mean when a hill has a grade of 15%? It means the slope is 15%, so it rises 15 feet over a horizontal distance of 100 feet. ut that can e hard to comprehend. Knowing the angle of elevation might give you a etter idea of how steep and difficult you can epect your clim to e. The first trigonometric ratio you will learn aout, tangent, will show you how you can find the angle of elevation of a hill with a given grade. Find the measure of an angle, given its tangent value. KEYWORDS adjacent side opposite side trigonometric ratio inverse tangent tangent Trigonometric Ratios When you find the ratio of the lengths of two sides of a right triangle, you are finding a trigonometric ratio. These ratios relate the sides of the triangle to either of the acute angles. Specific to one of the acute angles, the sides are referred to as adjacent, opposite, and hypotenuse. The hypotenuse is the side opposite the right angle. The opposite side is across from the given angle, and the adjacent side is net to the given angle, ut is not the hypotenuse. a c For : With respect to, a is the opposite side, and is the adjacent side. With respect to, is the opposite side, and a is the adjacent side. c is always the hypotenuse. Tangents 289
4 The Tangent Ratio One of the most commonly used ratios is the tangent ratio. It is areviated tan. The tangent of an angle is the ratio of the length of the leg opposite that angle to the length of the leg adjacent to that angle. a c tan = opposite adjacent = a _ opposite tan = adjacent = a The tangent ratio in trigonometry is related to tangents to circles. Look at circle. It has a radius of 1. _ is tangent to circle at point. The tangent of is the ratio of the length of the tangent segment to the radius. 1 _ opposite tan = adjacent = _ = _ 1 = That is, the tangent of the angle is the length of the tangent segment. Trigonometric ratios can e epressed as either a fraction or decimal. E F 5 D tan D = _ 12 5 = 2.4 tan E = _ You can use a calculator to find a decimal approimation for a tangent ratio. To do so, set your calculator in degree mode and use the tan key. The screen elow shows that the tangent of a 55 angle is approimately To find the value of in the diagram elow, we set up the tangent ratio and solve for tan 43 = _ (tan 43 ) = Unit 10 TRIGOnometry
5 The Inverse Tangent Sometimes you know the lengths of the legs of a right triangle and you want to find the measure of an acute angle. For this, you use the inverse tangent, areviated tan 1. Your calculator may have tan 1, or you may have to use an INV key. s shown elow, if the tangent of an angle is 5 3, or 0.6, the angle measure is approimately 31. The grade of a hill is given to e 15%. To find the angle of elevation, draw a right triangle with a slope of Then find the inverse tangent tan = tan = The angle of elevation is aout 8.5. Summary The tangent of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: tan = _ opposite adjacent. You use the tangent of an angle to find the length of a missing leg in a right triangle. To find a missing angle, use the inverse tangent: tan 1. Tangents 291
6
7 Sines and osines Ojectives Epress the sine and cosine ratios of given angles as fractions or decimals. Use a calculator to find the value of the sine or cosine of an angle. Use the inverse sine and cosine to find the measure of an angle. Trigonometric ratios can e used to make indirect measurements. For instance, suppose you are flying a kite and let it out to its full length. y knowing the length of the string and y estimating the angle the string is making with the ground, you can estimate how far directly aove the ground the kite is flying. KEYWORDS cosine identity sine The Sine and osine Ratios The tangent ratio only involves the legs of a right triangle, ecause opposite and adjacent (the two legs) are never the hypotenuse. Now you will learn aout two more trigonometric ratios, oth of which involve the hypotenuse. They are sine and cosine, areviated sin and cos. The sine of an angle is the ratio of the length of the leg that is opposite the angle to the length of the hypotenuse. a c opposite sin = hypotenuse = a c sin = opposite hypotenuse = c Sines and osines 293
8 The cosine of an angle is the ratio of the length of the leg adjacent to that angle to the length of the hypotenuse. a c adjacent cos = hypotenuse = c adjacent cos = hypotenuse = a c D 1 E F D 1 sin tan cos E F s with tangent, oth sine and cosine can e defined as lengths of segments on a circle with a radius of 1. Return to circle where _ is a tangent segment,, D, and are collinear, and, E, and are also collinear. _ DE is a half-chord ecause _ is a perpendicular isector of chord _ DF. The original word for half-chord in Sanskrit was translated to raic, and then mistranslated to the Latin word sinus, which we now call sine. opposite sin = hypotenuse = _ DE D = _ DE 1 = DE adjacent cos = hypotenuse = _ E D = _ E 1 = E The illustration at left summarizes how the ratios of sine, cosine, and tangent relate to the circle. lso like the tangent ratio, the sine and cosine ratios can e epressed as either fractions or decimals. E 8 10 F 6 D sin D = _ 8 10 = 0.8 sin E = _ 6 10 = 0.6 _ 6 cos D = 10 = 0.6 cos E = _ 8 10 = 0.8 Notice that the sine of one acute angle is the same as the cosine of the other acute angle from the same triangle. s you did for the tangent ratio, you can use your calculator to find decimal approimations for the sine and cosine of an angle. Use the sin and cos keys on your calculator. The screen elow shows that the sine of a 35 angle is approimately and the cosine is approimately Rememer that your calculator must e set in degree mode. 294 Unit 10 TRIGOnometry
9 Here is an eample of how to use sine to find a missing side length _ sin 61 = (sin 61 ) = nd here is an eample that uses cosine y _ 10 cos 36 = y y(cos 36 ) = 10 _ 10 y = cos 36 y You can use trigonometry to make indirect measurements. Suppose a kite is at the end of a 250 ft string. If the string makes an angle of aout 50 with the ground, we can approimate the height of the kite aove the ground y setting up and solving the following equation sin 50 = (sin 50 ) = The kite is aout 192 feet high. The Inverse Sine and osine To find the measure of an acute angle, given the length of a leg and the length of the hypotenuse, you use the inverse sine and cosine: sin 1 and cos 1. s shown elow, if the sine of an angle is 1 2, or 0.5, the angle measure is 30. If the cosine of an angle is 0.5, the angle measure is 60. Sines and osines 295
10 16-foot ladder is leaning against a house. The top of the ladder reaches 10 feet aove the ground. To find the angle the ase of the ladder makes with the ground, we set up an equation and find the inverse sine _ 10 sin = 16 sin = Trigonometric Identities trigonometric identity is an equation containing a trigonometric ratio that is true for all values of the variale. There are many trigonometric identities. We will eamine two of them and show they are true. _ sin Identity 1 tan = cos MemoryTip To rememer that sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent, use each first letter to make the word sohcahtoa, pronounced soak-a-toe-ah. To show that this is true, we sustitute the ratios for sine and cosine into the equation. opposite hypotenuse tan = adjacent hypotenuse opposite tan = hypotenuse hypotenuse adjacent tan = _ opposite adjacent The last equation is the definition of the tangent ratio, so the original equation is true. Identity 2 sin 2 + cos 2 = 1 y (, y) We can use a circle on the coordinate plane to show that this identity is true. Its center is located at (0, 0) and it has a radius of 1. Since all radii of a circle are congruent, the hypotenuse of the triangle has a radius of 1, and we can write the sine and cosine of angle a as follows: a y (1, 0) y sin a = 1 = y and cos a = 1 = The equation of a circle with its center at the origin and a radius 1 is: ( 0) 2 + (y 0) 2 = y 2 = 1 Now we use the ommutative Property of ddition and sustitute sin a and cos a into the equation for and y. (sin a) 2 + (cos a) 2 = Unit 10 TRIGOnometry
11 To make things a it easier to read, mathematicians usually replace (sin a) 2 with sin 2 a, so the identity can e written in the form sin 2 a + cos 2 a = 1 Summary The sine of an angle is the ratio of the length of the side opposite the angle opposite to the length of the hypotenuse: sin = hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent the adjacent angle to the length of the hypotenuse: cos = hypotenuse. You can use the sine and cosine of an angle to find a missing side of a right triangle. To find a missing angle, you can use the inverses: sin 1 and cos 1. trigonometric identity is an equation containing a trigonometric ratio that is true for all values of the variale. Two trigonometric identities are: tan = _ cos sin and sin2 + cos 2 = 1. Sines and osines 297
12
13 Special Right Triangles Ojectives Use trigonometric definitions and special right triangles to derive trigonometric ratios for 30, 45, and 60 angles. Solve prolems involving special right triangles y using trigonometric ratios. In life, there are often many ways to solve a prolem. It s the same with mathematics. For some triangles, you can find a missing side y applying the Pythagorean Theorem, y using a formula, or y writing a proportion. They all give the same answer. So, how do you know which way to go? The choice is yours. The decision may depend on whether or not you have a calculator. Or you can solve a prolem one way, and use another way to check your answer. The Triangle ecause of the measures of its angles, the triangle is commonly used, so knowing the trigonometric ratios for a 45 angle can come in handy. Let us first review the relationship etween the legs and hypotenuse of a triangle. When you construct the diagonal of a square, you create two triangles with a side length of s. You can then use the Pythagorean Theorem to see that the length of the diagonal is 2 times the length of a side. d = s 2 45 s 45 s s 2 + s 2 = d 2 2s 2 = d 2 2s 2 = d 2 s 2 = d Rememer radical epression is not considered simplified when the square root is in the denominator. t right, we rationalized the denominator y multiplying _ 1 y _ Now use s and s 2 to find the sine, cosine, and tangent of a 45 angle. You can use either of the 45 angles. opposite sin 45 = hypotenuse = s _ = 1 = _ 2 s adjacent cos 45 = hypotenuse = s _ = 1 = _ 2 s _ opposite tan 45 = adjacent = s_ s = 1 SPEIL RIGHT TRINGLES 299
14 The Triangle The triangle is also a commonly used triangle, so it is a good idea to know the trigonometric ratios for 30 and 60. When you construct the altitude of an equilateral triangle, you create two triangles with a hypotenuse of s and a ase (the shorter side) of s 2. gain, you can use the Pythagorean Theorem to see that the height is 3 times the length of the shorter side. _ h = s s 60 s 2 ( s 2 ) 2 + h 2 = s 2 h 2 = s 2 s2 4 h 2 _ = 4s2 4 s 2 4 h 2 _ = 3s2 4 s h = 3 2 s Now use s, 2, and s 3 2 to find the sine, cosine, and tangent of a 30 and 60 angle. s opposite sin 30 = hypotenuse = 2 s = s 2s = 2 1 _ opposite sin 60 = hypotenuse = s 3 _ 2 s = _ s 3 2s = 3 2 _ cos 30 = adjacent hypotenuse = s 3 _ 2 s = _ s 3 2s = _ 3 2 cos 60 = adjacent s hypotenuse = 2 s = s 2s = 2 1 _ tan 30 = opposite s adjacent = 2 _ 2s = = _ 1 = _ 3 s 3 2 2s _ tan 60 = opposite adjacent = s 3 2 s 2 = _ 2s 3 2s = Unit 10 TRIGOnometry
15 Using the Trigonometric Ratios to Solve Prolems construction worker stands on a rooftop 65 feet aove the ground. She throws down a rope, which another worker should anchor to the ground at a 30 angle. To the nearest foot, how far away from the ase of the uilding should the rope e anchored to make the correct angle? Use the tangent of a 30 angle to set up and solve a proportion. _ 65 tan 30 = 65 _ = _ 65 3 = The rope should e anchored 113 feet from the ase of the uilding. display screen is shaped like a square with a diagonal length of 14 centimeters. To the nearest tenth of a centimeter, find the length and width of the screen. Use the sine (or cosine) of a 45 angle to set up and solve a proportion. 14 cm _ sin 45 = 14 _ 2 2 = _ 14 2 = 14 2 = The screen has a length and width of aout 9.9 centimeters. SPEIL RIGHT TRINGLES 301
16 Summary You can find the sine, cosine, and tangent of a 45 angle y applying the trigonometric definitions to the sides of a triangle. You can find the sine, cosine, and tangent of a 30 angle and a 60 angle y applying the trigonometric definitions to the sides of a triangle. The trigonometric ratios that can e derived from special right triangles are summarized in the tale. ngle Measure sine cosine tangent _ 3 2 _ _ 2 _ _ You can find the lengths of missing sides in special right triangles y using trigonometric ratios. 302 Unit 10 TRIGOnometry
17 The Laws of Sines and osines Ojectives Recognize when to use the Law of Sines or the Law of osines. Use the Laws of Sines and osines to find missing measures in triangles. The Pythagorean Theorem is an invaluale tool, ut it has its limitations. It is only true for right triangles, and there are many situations that involve triangles that are not right triangles. For eample, imagine that two oats leave a dock at the same time and head in different directions. t any point in time, the two oats and dock make up the vertices of a triangle. The new formulas you will learn will allow you to find side and angle measures for any type of triangle, not just a right triangle. KEYWORD solve a triangle The Law of Sines So far we have used trigonometric ratios only to solve right triangles. We can also use trigonometry to solve triangles that are not right. One way is to use the Law of Sines. We will prove it efore we state it. onsider elow. It is not a right triangle, ut y drawing the altitude _ D, we have created two right triangles: D and D. h a D h First we find the sine of : sin = and solve for h: h = sin Then we find the sine of : h sin = a and solve for h: Now we have two epressions equal to h. y sustitution: Finally, we divide oth sides y a: h = a sin sin = a sin _ sin a = _ sin The Laws of Sines and osines 303
18 y drawing different altitudes, we can show that _ sin a = _ sin = _ sin c. This is the Law of Sines. The Law of Sines c a For any, where a,, and c are the measures of the opposite sides of,, and, respectively, _ sin a = _ sin = _ sin c Using the Law of Sines, you can find a missing triangle measure in the following two cases. You are given two angles and any side (S or S). You are given two sides and the nonincluded angle (SS). When you use the Law of Sines, you set up and solve a proportion using two of the three ratios. To find the length of side a elow, we set up a proportion using the ratios with and. Rememer The largest angle is across from the largest side and the smallest angle is across from the smallest side. For the prolem at right, is the greatest angle, so a must e less than a _ sin a = _ sin sin 55 a = sin a(sin 75 ) = 20(sin 55 ) 20(sin 55 ) a = sin 75 a 17 You can also use the Law of Sines to find angle measures of triangles. To find the measure of E elow, we set up a proportion and use the inverse sine. F 7 10 E 44 D _ sin E e = _ sin D d _ sin E 10 = sin (sin E) = 10(sin 44 ) 10(sin 44 ) sin E = 7 10(sin 44 ) m E = sin ( 1 7 ) Unit 10 TRIGOnometry
19 The Law of osines The Law of Sines can only e used for certain angle and side cominations. When you cannot use the Law of Sines, you may e ale to use the Law of osines. We will prove the law efore we state it. gain, we have with altitude _ D creating two right triangles: D and D. Let D = and D = c. D h c a c First we find the cosine of angle : cos = and we solve for : = cos We use the Pythagorean Theorem for D: 2 = h and solve for h 2 : h 2 = 2 2 Then we use the Pythagorean Theorem in D: a 2 = h 2 + (c ) 2 and square the inomial: a 2 = h 2 + c 2 2c + 2 Now we sustitute for h 2 : a 2 = c 2 2c + 2 comine like terms: a 2 = 2 + c 2 2c sustitute for : a 2 = 2 + c 2 2c( cos ) and rearrange factors: a 2 = 2 + c 2 2c cos We can draw different altitudes and follow similar reasoning for each. The Law of osines c a For any, where a,, and c are the measures of the opposite sides of,, and, respectively, a 2 = 2 + c 2 2c cos 2 = a 2 + c 2 2ac cos c 2 = a a cos Use the Law of osines to find a missing triangle measure in the following two cases. You are given two sides and the included angle (SS). You are given three sides (SSS). When you use the Law of osines to find a missing side, choose the formula that has that side on the left side of the equation. To find the length of side elow, we choose the formula with 2 on the left side = a 2 + c 2 2ac cos 2 = (4)(10) cos 34 2 = cos The Laws of Sines and osines 305
20 If you know the lengths of three sides of a triangle, you can use the Law of osines to find the measure of any angle of the triangle. Two ships leave a dock at the same time and head in different directions without changing course. fter a few hours, one ship is 200 miles from the dock and the other is 325 miles from the dock. They are 410 miles from each other. What angle is formed y the paths of the two ships at the dock? In the diagram, is the missing angle, so we use the formula that has on the right side. Notice elow that we isolate cos and then use the inverse cosine. 410 mi 325 mi 200 mi a 2 = 2 + c 2 2c cos = (200)(325)cos 168,100 = 145, ,000 cos 22,475 = 130,000 cos _ 22, ,000 = cos m 100 If you want to solve the triangle, which means to find all missing measures, you can now use the Law of Sines to find another angle, and then sutract the sum of the two known measures from 180 to find the last angle measure. Summary The Laws of Sines and osines can e used to solve triangles other than right triangles. The Law of Sines states that for any, _ sin a = _ sin = _ sin c. Use the Law of Sines when you are given two angles and any side, or two sides and a nonincluded angle. The Law of osines states that for any, a 2 = 2 + c 2 2c cos 2 = a 2 + c 2 2ac cos c 2 = a a cos Use the Law of osines when you are given two sides and the included angle, or all three sides. 306 Unit 10 TRIGOnometry
Be sure to label all answers and leave answers in exact simplified form.
Pythagorean Theorem word problems Solve each of the following. Please draw a picture and use the Pythagorean Theorem to solve. Be sure to label all answers and leave answers in exact simplified form. 1.
More informationHistorical Note Trigonometry Ratios via Similarity
Section 12-6 Trigonometry Ratios via Similarity 1 12-6 Trigonometry Ratios via Similarity h 40 190 ft of elevation Figure 12-83 Measurements of buildings, structures, and some other objects are frequently
More informationName: Block: What I can do for this unit:
Unit 8: Trigonometry Student Tracking Sheet Math 10 Common Name: Block: What I can do for this unit: After Practice After Review How I Did 8-1 I can use and understand triangle similarity and the Pythagorean
More informationChapter 7. Right Triangles and Trigonometry
hapter 7 Right Triangles and Trigonometry 7.1 pply the Pythagorean Theorem 7.2 Use the onverse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 pply the Tangent
More informationAW Math 10 UNIT 7 RIGHT ANGLE TRIANGLES
AW Math 10 UNIT 7 RIGHT ANGLE TRIANGLES Assignment Title Work to complete Complete 1 Triangles Labelling Triangles 2 Pythagorean Theorem 3 More Pythagorean Theorem Eploring Pythagorean Theorem Using Pythagorean
More informationGeometry- Unit 6 Notes. Simplifying Radicals
Geometry- Unit 6 Notes Name: Review: Evaluate the following WITHOUT a calculator. a) 2 2 b) 3 2 c) 4 2 d) 5 2 e) 6 2 f) 7 2 g) 8 2 h) 9 2 i) 10 2 j) 2 2 k) ( 2) 2 l) 2 0 Simplifying Radicals n r Example
More information7.1/7.2 Apply the Pythagorean Theorem and its Converse
7.1/7.2 Apply the Pythagorean Theorem and its Converse Remember what we know about a right triangle: In a right triangle, the square of the length of the is equal to the sum of the squares of the lengths
More informationAccel. Geometry - Concepts Similar Figures, Right Triangles, Trigonometry
Accel. Geometry - Concepts 16-19 Similar Figures, Right Triangles, Trigonometry Concept 16 Ratios and Proportions (Section 7.1) Ratio: Proportion: Cross-Products Property If a b = c, then. d Properties
More informationYou ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46
Math 1330 Section 6.2 Section 7.1: Right-Triangle Applications In this section, we ll solve right triangles. In some problems you will be asked to find one or two specific pieces of information, but often
More informationAWM 11 UNIT 4 TRIGONOMETRY OF RIGHT TRIANGLES
AWM 11 UNIT 4 TRIGONOMETRY OF RIGHT TRIANGLES Assignment Title Work to complete Complete 1 Triangles Labelling Triangles 2 Pythagorean Theorem Exploring Pythagorean Theorem 3 More Pythagorean Theorem Using
More information14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio.
14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio. Using the space below, draw at least right triangles, each of which has one
More informationName Class Date. Investigating a Ratio in a Right Triangle
Name lass Date Trigonometric Ratios Going Deeper Essential question: How do you find the tangent, sine, and cosine ratios for acute angles in a right triangle? In this chapter, you will be working etensively
More informationAssignment Guide: Chapter 8 Geometry (L3)
Assignment Guide: Chapter 8 Geometry (L3) (91) 8.1 The Pythagorean Theorem and Its Converse Page 495-497 #7-31 odd, 37-47 odd (92) 8.2 Special Right Triangles Page 503-504 #7-12, 15-20, 23-28 (93) 8.2
More information(13) Page #1 8, 12, 13, 15, 16, Even, 29 32, 39 44
Geometry/Trigonometry Unit 7: Right Triangle Notes Name: Date: Period: # (1) Page 430 #1 15 (2) Page 430 431 #16 23, 25 27, 29 and 31 (3) Page 437 438 #1 8, 9 19 odd (4) Page 437 439 #10 20 Even, 23, and
More informationAssignment. Pg. 567 #16-33, even pg 577 # 1-17 odd, 32-37
Assignment Intro to Ch. 8 8.1 8. Da 1 8. Da 8. Da 1 8. Da Review Quiz 8. Da 1 8. Da 8. Etra Practice 8.5 8.5 In-class project 8.6 Da 1 8.6 Da Ch. 8 review Worksheet Worksheet Worksheet Worksheet Worksheet
More informationhypotenuse adjacent leg Preliminary Information: SOH CAH TOA is an acronym to represent the following three 28 m 28 m opposite leg 13 m
On Twitter: twitter.com/engagingmath On FaceBook: www.mathworksheetsgo.com/facebook I. odel Problems II. Practice Problems III. Challenge Problems IV. Answer ey Web Resources Using the inverse sine, cosine,
More informationName Class Date. Essential question: How do you find the tangent, sine, and cosine ratios for acute angles in a right triangle?
Name lass Date 8-2 Trigonometric Ratios Going Deeper Essential question: How do you find the tangent, sine, and cosine ratios for acute angles in a right triangle? In this chapter, you will be working
More informationInequalities in Triangles Geometry 5-5
Inequalities in Triangles Geometry 5-5 Name: ate: Period: Theorem 5-10 Theorem 5-11 If two sides of a triangle are not If two angles of a triangle are not congruent, then the larger angle congruent, then
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
Summer Review for Students Entering Pre-Calculus with Trigonometry 1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios
More informationCumulative Review: SOHCAHTOA and Angles of Elevation and Depression
Cumulative Review: SOHCAHTOA and Angles of Elevation and Depression Part 1: Model Problems The purpose of this worksheet is to provide students the opportunity to review the following topics in right triangle
More informationabout touching on a topic and then veering off to talk about something completely unrelated.
The Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent 8.2 Learning Goals In this lesson, you will: Use the tangent ratio in a right triangle to solve for unknown side lengths. Use the cotangent
More informationAlgebra II. Slide 1 / 92. Slide 2 / 92. Slide 3 / 92. Trigonometry of the Triangle. Trig Functions
Slide 1 / 92 Algebra II Slide 2 / 92 Trigonometry of the Triangle 2015-04-21 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 92 Trigonometry of the Right Triangle Inverse
More informationUnit 6: Triangle Geometry
Unit 6: Triangle Geometry Student Tracking Sheet Math 9 Principles Name: lock: What I can do for this unit: fter Practice fter Review How I id 6-1 I can recognize similar triangles using the ngle Test,
More informationA lg e b ra II. Trig o n o m e try o f th e Tria n g le
1 A lg e b ra II Trig o n o m e try o f th e Tria n g le 2015-04-21 www.njctl.org 2 Trig Functions click on the topic to go to that section Trigonometry of the Right Triangle Inverse Trig Functions Problem
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios and Pythagorean Theorem 4. Multiplying and Dividing Rational Expressions
More informationUNIT 5 TRIGONOMETRY Lesson 5.4: Calculating Sine, Cosine, and Tangent. Instruction. Guided Practice 5.4. Example 1
Lesson : Calculating Sine, Cosine, and Tangent Guided Practice Example 1 Leo is building a concrete pathway 150 feet long across a rectangular courtyard, as shown in the following figure. What is the length
More informationIntroduction to Trigonometry
NAME COMMON CORE GEOMETRY- Unit 6 Introduction to Trigonometry DATE PAGE TOPIC HOMEWORK 1/22 2-4 Lesson 1 : Incredibly Useful Ratios Homework Worksheet 1/23 5-6 LESSON 2: Using Trigonometry to find missing
More informationApply the Tangent Ratio. You used congruent or similar triangles for indirect measurement. You will use the tangent ratio for indirect measurement.
7.5 pply the Tangent Ratio efore Now You used congruent or similar triangles for indirect measurement. You will use the tangent ratio for indirect measurement. Why? So you can find the height of a roller
More informationThe three primary Trigonometric Ratios are Sine, Cosine, and Tangent. opposite. Find sin x, cos x, and tan x in the right triangles below:
Trigonometry Geometry 12.1 The three primary Trigonometric Ratios are Sine, osine, and Tangent. s we learned previously, triangles with the same angle measures have proportional sides. If you know one
More informationObjectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using
Ch 13 - RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric
More information2.0 Trigonometry Review Date: Pythagorean Theorem: where c is always the.
2.0 Trigonometry Review Date: Key Ideas: The three angles in a triangle sum to. Pythagorean Theorem: where c is always the. In trigonometry problems, all vertices (corners or angles) of the triangle are
More informationSolving Right Triangles. How do you solve right triangles?
Solving Right Triangles How do you solve right triangles? The Trigonometric Functions we will be looking at SINE COSINE TANGENT The Trigonometric Functions SINE COSINE TANGENT SINE Pronounced sign TANGENT
More informationName: Unit 8 Right Triangles and Trigonometry Unit 8 Similarity and Trigonometry. Date Target Assignment Done!
Unit 8 Similarity and Trigonometry Date Target Assignment Done! M 1-22 8.1a 8.1a Worksheet T 1-23 8.1b 8.1b Worksheet W 1-24 8.2a 8.2a Worksheet R 1-25 8.2b 8.2b Worksheet F 1-26 Quiz Quiz 8.1-8.2 M 1-29
More informationNon-right Triangles: Law of Cosines *
OpenStax-CNX module: m49405 1 Non-right Triangles: Law of Cosines * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:
More informationCh 8: Right Triangles and Trigonometry 8-1 The Pythagorean Theorem and Its Converse 8-2 Special Right Triangles 8-3 The Tangent Ratio
Ch 8: Right Triangles and Trigonometry 8-1 The Pythagorean Theorem and Its Converse 8- Special Right Triangles 8-3 The Tangent Ratio 8-1: The Pythagorean Theorem and Its Converse Focused Learning Target:
More information10-2. More Right-Triangle Trigonometry. Vocabulary. Finding an Angle from a Trigonometric Ratio. Lesson
hapter 10 Lesson 10-2 More Right-Triangle Trigonometry IG IDE If you know two sides of a right triangle, you can use inverse trigonometric functions to fi nd the measures of the acute angles. Vocabulary
More information8.4 Special Right Triangles
8.4. Special Right Triangles www.ck1.org 8.4 Special Right Triangles Learning Objectives Identify and use the ratios involved with isosceles right triangles. Identify and use the ratios involved with 30-60-90
More informationSM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or
SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length
More informationAdvanced Math Final Exam Review Name: Bornoty May June Use the following schedule to complete the final exam review.
Advanced Math Final Exam Review Name: Bornoty May June 2013 Use the following schedule to complete the final exam review. Homework will e checked in every day. Late work will NOT e accepted. Homework answers
More information10-1. Three Trigonometric Functions. Vocabulary. Lesson
Chapter 10 Lesson 10-1 Three Trigonometric Functions BIG IDEA The sine, cosine, and tangent of an acute angle are each a ratio of particular sides of a right triangle with that acute angle. Vocabulary
More informationG.8 Right Triangles STUDY GUIDE
G.8 Right Triangles STUDY GUIDE Name Date Block Chapter 7 Right Triangles Review and Study Guide Things to Know (use your notes, homework, quizzes, textbook as well as flashcards at quizlet.com (http://quizlet.com/4216735/geometry-chapter-7-right-triangles-flashcardsflash-cards/)).
More informationSolving Right Triangles. LEARN ABOUT the Math
7.5 Solving Right Triangles GOL Use primary trigonometric ratios to calculate side lengths and angle measures in right triangles. LERN OUT the Math farmers co-operative wants to buy and install a grain
More informationBenchmark Test 4. Pythagorean Theorem. More Copy if needed. Answers. Geometry Benchmark Tests
enchmark LESSON 00.00 Tests More opy if needed enchmark Test 4 Pythagorean Theorem 1. What is the length of the hypotenuse of a right triangle with leg lengths of 12 and 6?. 3 Ï } 2. Ï } 144. 6 Ï } 3 D.
More informationGeometry SIA #3. Name: Class: Date: Short Answer. 1. Find the perimeter of parallelogram ABCD with vertices A( 2, 2), B(4, 2), C( 6, 1), and D(0, 1).
Name: Class: Date: ID: A Geometry SIA #3 Short Answer 1. Find the perimeter of parallelogram ABCD with vertices A( 2, 2), B(4, 2), C( 6, 1), and D(0, 1). 2. If the perimeter of a square is 72 inches, what
More informationChapter 7 - Trigonometry
Chapter 7 Notes Lessons 7.1 7.5 Geometry 1 Chapter 7 - Trigonometry Table of Contents (you can click on the links to go directly to the lesson you want). Lesson Pages 7.1 and 7.2 - Trigonometry asics Pages
More informationChapter 7: Right Triangles and Trigonometry Name: Study Guide Block: Section and Objectives
Page 1 of 22 hapter 7: Right Triangles and Trigonometr Name: Stud Guide lock: 1 2 3 4 5 6 7 8 SOL G.8 The student will solve real-world problems involving right triangles b using the Pthagorean Theorem
More informationYou ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46
Math 1330 Section 6.2 Section 7.1: Right-Triangle Applications In this section, we ll solve right triangles. In some problems you will be asked to find one or two specific pieces of information, but often
More informationBe sure to label all answers and leave answers in exact simplified form.
Pythagorean Theorem word problems Solve each of the following. Please draw a picture and use the Pythagorean Theorem to solve. Be sure to label all answers and leave answers in exact simplified form. 1.
More informationAngles of a Triangle. Activity: Show proof that the sum of the angles of a triangle add up to Finding the third angle of a triangle
Angles of a Triangle Activity: Show proof that the sum of the angles of a triangle add up to 180 0 Finding the third angle of a triangle Pythagorean Theorem Is defined as the square of the length of the
More informationUNIT 9 - RIGHT TRIANGLES AND TRIG FUNCTIONS
UNIT 9 - RIGHT TRIANGLES AND TRIG FUNCTIONS Converse of the Pythagorean Theorem Objectives: SWBAT use the converse of the Pythagorean Theorem to solve problems. SWBAT use side lengths to classify triangles
More informationGeometry First Semester Practice Final (cont)
49. Determine the width of the river, AE, if A. 6.6 yards. 10 yards C. 12.8 yards D. 15 yards Geometry First Semester Practice Final (cont) 50. In the similar triangles shown below, what is the value of
More informationUnit 1 Trigonometry. Topics and Assignments. General Outcome: Develop spatial sense and proportional reasoning. Specific Outcomes:
1 Unit 1 Trigonometry General Outcome: Develop spatial sense and proportional reasoning. Specific Outcomes: 1.1 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems
More information3.0 Trigonometry Review
3.0 Trigonometry Review In trigonometry problems, all vertices (corners or angles) of the triangle are labeled with capital letters. The right angle is usually labeled C. Sides are usually labeled with
More informationTheorem 8-1-1: The three altitudes in a right triangle will create three similar triangles
G.T. 7: state and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Understand and use the geometric mean to solve for missing parts of triangles. 8-1
More informationMath-2 Lesson 8-7: Unit 5 Review (Part -2)
Math- Lesson 8-7: Unit 5 Review (Part -) Trigonometric Functions sin cos A A SOH-CAH-TOA Some old horse caught another horse taking oats away. opposite ( length ) o sin A hypotenuse ( length ) h SOH adjacent
More information2 nd Semester Final Exam Review
2 nd Semester Final xam Review I. Vocabulary hapter 7 cross products proportion scale factor dilation ratio similar extremes scale similar polygons indirect measurements scale drawing similarity ratio
More informationLesson Title 2: Problem TK Solving with Trigonometric Ratios
Part UNIT RIGHT solving TRIANGLE equations TRIGONOMETRY and inequalities Lesson Title : Problem TK Solving with Trigonometric Ratios Georgia Performance Standards MMG: Students will define and apply sine,
More informationG r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S )
G r a d e 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 0 S ) Midterm Practice Exam Answer Key G r a d e 0 I n t r o d u c t i o n t o A p p l i e d
More informationPage 1. Right Triangles The Pythagorean Theorem Independent Practice
Name Date Page 1 Right Triangles The Pythagorean Theorem Independent Practice 1. Tony wants his white picket fence row to have ivy grow in a certain direction. He decides to run a metal wire diagonally
More informationI. Model Problems II. Practice III. Challenge Problems IV. Answer Key. Sine, Cosine Tangent
On Twitter: twitter.com/engagingmath On FaceBook: www.mathworksheetsgo.com/facebook I. Model Problems II. Practice III. Challenge Problems IV. Answer Key Web Resources Sine, Cosine Tangent www.mathwarehouse.com/trigonometry/sine-cosine-tangent.html
More informationLesson 2: Right Triangle Trigonometry
Lesson : Right Triangle Trigonometry lthough Trigonometry is used to solve many prolems, historially it was first applied to prolems that involve a right triangle. This an e extended to non-right triangles
More informationTrigonometric Ratios and Functions
Algebra 2/Trig Unit 8 Notes Packet Name: Date: Period: # Trigonometric Ratios and Functions (1) Worksheet (Pythagorean Theorem and Special Right Triangles) (2) Worksheet (Special Right Triangles) (3) Page
More informationFinding Angles and Solving Right Triangles NEW SKILLS: WORKING WITH INVERSE TRIGONOMETRIC RATIOS. Calculate each angle to the nearest degree.
324 MathWorks 10 Workbook 7.5 Finding Angles and Solving Right Triangles NEW SKILLS: WORKING WITH INVERSE TRIGONOMETRIC RATIOS The trigonometric ratios discussed in this chapter are unaffected by the size
More informationGeometry Unit 3 Practice
Lesson 17-1 1. Find the image of each point after the transformation (x, y) 2 x y 3, 3. 2 a. (6, 6) b. (12, 20) Geometry Unit 3 ractice 3. Triangle X(1, 6), Y(, 22), Z(2, 21) is mapped onto XʹYʹZʹ by a
More informationReady To Go On? Skills Intervention 8-1 Similarity in Right Triangles
8 Find this vocabular word in Lesson 8-1 and the Multilingual Glossar. Finding Geometric Means The geometric mean of two positive numbers is the positive square root of their. Find the geometric mean of
More informationReview of Sine, Cosine, and Tangent for Right Triangle
Review of Sine, Cosine, and Tangent for Right Triangle In trigonometry problems, all vertices (corners or angles) of the triangle are labeled with capital letters. The right angle is usually labeled C.
More informationTriangle Trigonometry
Honors Finite/Brief: Trigonometry review notes packet Triangle Trigonometry Right Triangles All triangles (including non-right triangles) Law of Sines: a b c sin A sin B sin C Law of Cosines: a b c bccos
More informationAreas of Rectangles and Parallelograms
CONDENSED LESSON 8.1 Areas of Rectangles and Parallelograms In this lesson, you Review the formula for the area of a rectangle Use the area formula for rectangles to find areas of other shapes Discover
More information1. The Pythagorean Theorem
. The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be
More informationStudy Guide and Review
Choose the term that best matches the statement or phrase. a square of a whole number A perfect square is a square of a whole number. a triangle with no congruent sides A scalene triangle has no congruent
More informationPractice For use with pages
9.1 For use with pages 453 457 Find the square roots of the number. 1. 36. 361 3. 79 4. 1089 5. 4900 6. 10,000 Approimate the square root to the nearest integer. 7. 39 8. 85 9. 105 10. 136 11. 17.4 1.
More informationPart Five: Trigonometry Review. Trigonometry Review
T.5 Trigonometry Review Many of the basic applications of physics, both to mechanical systems and to the properties of the human body, require a thorough knowledge of the basic properties of right triangles,
More informationGeometry: Traditional Pathway
GEOMETRY: CONGRUENCE G.CO Prove geometric theorems. Focus on validity of underlying reasoning while using variety of ways of writing proofs. G.CO.11 Prove theorems about parallelograms. Theorems include:
More informationGEOMETRY. Background Knowledge/Prior Skills. Knows ab = a b. b =
GEOMETRY Numbers and Operations Standard: 1 Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers, and number
More informationChapter Nine Notes SN P U1C9
Chapter Nine Notes SN P UC9 Name Period Section 9.: Applications Involving Right Triangles To evaluate trigonometric functions with a calculator, there are a few important things to know: On your calculator,
More informationSemester Exam Review. Honors Geometry A
Honors Geometry 2015-2016 The following formulas will be provided in the student examination booklet. Pythagorean Theorem In right triangle with right angle at point : 2 2 2 a b c b c a Trigonometry In
More informationSection 10.6 Right Triangle Trigonometry
153 Section 10.6 Right Triangle Trigonometry Objective #1: Understanding djacent, Hypotenuse, and Opposite sides of an acute angle in a right triangle. In a right triangle, the otenuse is always the longest
More informationThe Real Number System and Pythagorean Theorem Unit 9 Part C
The Real Number System and Pythagorean Theorem Unit 9 Part C Standards: 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;
More informationSkills Practice Skills Practice for Lesson 7.1
Skills Practice Skills Practice for Lesson.1 Name Date Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent Vocabulary Match each description to its corresponding term for triangle EFG. F
More information2.10 Theorem of Pythagoras
2.10 Theorem of Pythagoras Dr. Robert J. Rapalje, Retired Central Florida, USA Before introducing the Theorem of Pythagoras, we begin with some perfect square equations. Perfect square equations (see the
More informationUnit No: F3HW 11. Unit Title: Maths Craft 2. 4 Trigonometry Sine and Cosine Rules. Engineering and Construction
Unit No: F3HW 11 Unit Title: Maths Craft 4 Trigonometry Sine and Cosine Rules SINE AND COSINE RULES TRIGONOMETRIC RATIOS Remember: The word SOH CAH TOA is a helpful reminder. In any right-angled triangle,
More information5B.4 ~ Calculating Sine, Cosine, Tangent, Cosecant, Secant and Cotangent WB: Pgs :1-10 Pgs : 1-7
SECONDARY 2 HONORS ~ UNIT 5B (Similarity, Right Triangle Trigonometry, and Proof) Assignments from your Student Workbook are labeled WB Those from your hardbound Student Resource Book are labeled RB. Do
More informationGeo, Chap 8 Practice Test, EV Ver 1
Name: Class: Date: ID: A Geo, Chap 8 Practice Test, EV Ver 1 Short Answer Find the length of the missing side. Leave your answer in simplest radical form. 1. (8-1) 2. (8-1) A grid shows the positions of
More informationChapter 15 Right Triangle Trigonometry
Chapter 15 Right Triangle Trigonometry Sec. 1 Right Triangle Trigonometry The most difficult part of Trigonometry is spelling it. Once we get by that, the rest is a piece of cake. efore we start naming
More informationThe Sine of Things to Come Lesson 22-1 Similar Right Triangles
The Sine of Things to ome Lesson 22-1 Similar Right Triangles Learning Targets: Find ratios of side lengths in similar right triangles. Given an acute angle of a right triangle, identify the opposite leg
More information8. T(3, 4) and W(2, 7) 9. C(5, 10) and D(6, -1)
Name: Period: Chapter 1: Essentials of Geometry In exercises 6-7, find the midpoint between the two points. 6. T(3, 9) and W(15, 5) 7. C(1, 4) and D(3, 2) In exercises 8-9, find the distance between the
More informationSection Congruence Through Constructions
Section 10.1 - Congruence Through Constructions Definitions: Similar ( ) objects have the same shape but not necessarily the same size. Congruent ( =) objects have the same size as well as the same shape.
More informationChapter 3: Right Triangle Trigonometry
10C Name: Chapter 3: Right Triangle Trigonometry 3.1 The Tangent Ratio Outcome : Develop and apply the tangent ratio to solve problems that involve right triangles. Definitions: Adjacent side: the side
More informationUnit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8)
Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Standards: Geom 19.0, Geom 20.0, Trig 7.0, Trig 8.0, Trig 12.0 Segerstrom High School -- Math Analysis Honors Name: Period:
More informationChapter 9: Right Triangle Trigonometry
Haberman MTH 11 Section I: The Trigonometric Functions Chapter 9: Right Triangle Trigonometry As we studied in Intro to the Trigonometric Functions: Part 1, if we put the same angle in the center of two
More informationMATH STUDENT BOOK. 12th Grade Unit 3
MTH STUDENT OOK 12th Grade Unit 3 MTH 1203 RIGHT TRINGLE TRIGONOMETRY INTRODUTION 3 1. SOLVING RIGHT TRINGLE LENGTHS OF SIDES NGLE MESURES 13 INDIRET MESURE 18 SELF TEST 1: SOLVING RIGHT TRINGLE 23 2.
More informationGeometry. AIR Study Guide
Geometry AIR Study Guide Table of Contents Topic Slide Formulas 3 5 Angles 6 Lines and Slope 7 Transformations 8 Constructions 9 10 Triangles 11 Congruency and Similarity 12 Right Triangles Only 13 Other
More informationMathematics Placement Assessment
Mathematics Placement Assessment Courage, Humility, and Largeness of Heart Oldfields School Thank you for taking the time to complete this form accurately prior to returning this mathematics placement
More informationTheta Circles & Polygons 2015 Answer Key 11. C 2. E 13. D 4. B 15. B 6. C 17. A 18. A 9. D 10. D 12. C 14. A 16. D
Theta Circles & Polygons 2015 Answer Key 1. C 2. E 3. D 4. B 5. B 6. C 7. A 8. A 9. D 10. D 11. C 12. C 13. D 14. A 15. B 16. D 17. A 18. A 19. A 20. B 21. B 22. C 23. A 24. C 25. C 26. A 27. C 28. A 29.
More informationMoore Catholic High School Math Department
Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during
More information13.2 Sine and Cosine Ratios
Name lass Date 13.2 Sine and osine Ratios Essential Question: How can you use the sine and cosine ratios, and their inverses, in calculations involving right triangles? Explore G.9. Determine the lengths
More informationFind sin R and sin S. Then find cos R and cos S. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.
Name Homework Packet 7.6 7.7 LESSON 7.6 For use with pages 473-480 AND LESSON 7.7 For use with pages 483 489 Find sin R and sin S. Then find cos R and cos S. Write each answer as a fraction and as a decimal.
More informationUnit Circle. Project Response Sheet
NAME: PROJECT ACTIVITY: Trigonometry TOPIC Unit Circle GOALS MATERIALS Explore Degree and Radian Measure Explore x- and y- coordinates on the Unit Circle Investigate Odd and Even functions Investigate
More informationIf AB = 36 and AC = 12, what is the length of AD?
Name: ate: 1. ship at sea heads directly toward a cliff on the shoreline. The accompanying diagram shows the top of the cliff,, sighted from two locations, and B, separated by distance S. If m = 30, m
More informationThree Angle Measure. Introduction to Trigonometry. LESSON 9.1 Assignment
LESSON.1 Assignment Name Date Three Angle Measure Introduction to Trigonometry 1. Analyze triangle A and triangle DEF. Use /A and /D as the reference angles. E 7.0 cm 10.5 cm A 35 10.0 cm D 35 15.0 cm
More information