Essential Question How can you verify a trigonometric identity?
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1 9.7 Using Trigonometric Identities Essential Question How can you verify a trigonometric identity? Writing a Trigonometric Identity Work with a partner. In the figure, the point (, y) is on a circle of radius c with center at the origin. a. Write an equation that relates a, b, and c. y (, y) b. Write epressions for the sine and cosine ratios of angle. c b c. Use the results from parts (a) and (b) to find the sum of sin and cos. What do you observe? a d. Complete the table to verify that the identity you wrote in part (c) is valid for angles (of your choice) in each of the four quadrants. QI QII QIII QIV sin cos sin + cos Writing Other Trigonometric Identities REASONING ABSTRACTLY To be proficient in math, you need to know and fleibly use different properties of operations and objects. Work with a partner. The trigonometric identity you derived in Eploration is called a Pythagorean identity. There are two other Pythagorean identities. To derive them, recall the four relationships: tan = sec = cot = csc = a. Divide each side of the Pythagorean identity you derived in Eploration by cos and simplify. What do you observe? b. Divide each side of the Pythagorean identity you derived in Eploration by sin and simplify. What do you observe? Communicate Your Answer 3. How can you verify a trigonometric identity?. Is = a trigonometric identity? Eplain your reasoning.. Give some eamples of trigonometric identities that are different than those in Eplorations and. Section 9.7 Using Trigonometric Identities 3
2 9.7 Lesson What You Will Learn Core Vocabulary trigonometric identity, p. Previous unit circle STUDY TIP Note that sin represents () and cos represents (). Use trigonometric identities to evaluate trigonometric functions and simplify trigonometric epressions. Verify trigonometric identities. Using Trigonometric Identities Recall that when an angle is in standard position with its terminal side intersecting the unit circle at (, y), then = and y =. Because (, y) is on a circle centered at the origin with radius, it follows that + y = and cos + sin =. The equation cos + sin = is true for any value of. A 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. In Section 9., you used reciprocal identities to find the values of the cosecant, secant, and cotangent functions. These and other fundamental trigonometric identities are listed below. Core Concept Fundamental Trigonometric Identities Reciprocal Identities csc = sec = Tangent and Cotangent Identities tan = cot = cot = tan Pythagorean Identities sin + cos = + tan = sec + cot = csc Cofunction Identities sin ( π ) = cos ( π ) = tan ( π ) = cot Negative Angle Identities sin( ) = cos( ) = tan( ) = tan y r = (, ) = (, y) In this section, you will use trigonometric identities to do the following. Evaluate trigonometric functions. Simplify trigonometric epressions. Verify other trigonometric identities. Chapter 9 Trigonometric Ratios and Functions
3 Finding Trigonometric Values Given that = and π < < π, find the values of the other five trigonometric functions of. Step Find. sin + cos = ( ) Write Pythagorean identity. + cos = Substitute for. cos = ( ) cos = 9 Subtract ( ) from each side. Simplify. = ± 3 Take square root of each side. = 3 Because is in Quadrant II, is negative. Step Find the values of the other four trigonometric functions of using the values of and. tan = = 3 csc = = = 3 = cot = 3 = sec = = 3 Simplifying Trigonometric Epressions Simplify (a) tan ( π ) and (b) sec tan + sec. = 3 = 3 a. tan ( π ) = cot Cofunction identity = ( )() Cotangent identity = Simplify. b. sec tan + sec = sec (sec ) + sec Pythagorean identity = sec 3 sec + sec Distributive Property = sec 3 Simplify. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Given that = 6 and 0 < < π, find the values of the other five trigonometric functions of. Simplify the epression.. sin cot sec 3. sin. tan csc sec Section 9.7 Using Trigonometric Identities
4 Verifying Trigonometric Identities You can use the fundamental identities from this chapter to verify new trigonometric identities. When verifying an identity, begin with the epression on one side. Use algebra and trigonometric properties to manipulate the epression until it is identical to the other side. Verify the identity sec sec = sin. sec sec Verifying a Trigonometric Identity = sec sec sec = ( sec ) Write as separate fractions. Simplify. = cos Reciprocal identity = sin Pythagorean identity Notice that verifying an identity is not the same as solving an equation. When verifying an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. So, you cannot use any properties of equality, such as adding the same quantity to each side of the equation. Verifying a Trigonometric Identity cos Verify the identity sec + tan = sin. LOOKING FOR STRUCTURE To verify the identity, you must introduce sin into the denominator. Multiply the numerator and the denominator by sin so you get an equivalent epression. sec + tan = cos + tan = cos + sin cos = + sin cos = + sin cos sin sin sin = cos ( sin ) cos = cos ( sin ) cos = sin Reciprocal identity Tangent identity Add fractions. Multiply by sin sin. Simplify numerator. Pythagorean identity Simplify. Monitoring Progress Verify the identity. Help in English and Spanish at BigIdeasMath.com 6 Chapter 9 Trigonometric Ratios and Functions. cot( ) = cot 6. csc ( sin ) = cot 7. cos csc tan = 8. (tan + )(cos ) = tan
5 9.7 Eercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check. WRITING Describe the difference between a trigonometric identity and a trigonometric equation.. WRITING Eplain how to use trigonometric identities to determine whether sec( ) = sec or sec( ) = sec. Monitoring Progress and Modeling with Mathematics In Eercises 3 0, find the values of the other five trigonometric functions of. (See Eample.) 3. = 3, 0 < < π. = 7 0, π < < 3π ERROR ANALYSIS In Eercises and, describe and correct the error in simplifying the epression.. sin = ( + cos ) = cos = cos. tan = 3 7, π < < π 6. cot =, π < < π 7. = 6, π < < 3π. tan csc = cos sin sin = cos sin 8. sec = 9, 3π < < π 9. cot = 3, 3π < < π 0. csc = 3, π < < 3π In Eercises 0, simplify the epression. (See Eample.). sin cot. ( + tan ) 3.. sin( ) cos( ) cos ( π ) csc. csc 7. cot 8. sin( ) cot cos 9. ( π ) + cos csc sec sin + cos 0. ( π ) + sec cos cot 6. sin ( π ) sec cos tan ( ) cos In Eercises 3 30, verify the identity. (See Eamples 3 and.) 3. sin csc =. tan csc =. cos ( π ) cot = cos 6. sin ( π ) tan = sin cos ( π ) + = 8. sin( ) + cos sin sin + + cos = csc sin = csc + cot cos( ) sin ( ) tan = cos 3. USING STRUCTURE A function f is odd when f ( ) = f(). A function f is even when f ( ) = f (). Which of the si trigonometric functions are odd? Which are even? Justify your answers using identities and graphs. 3. ANALYZING RELATIONSHIPS As the value of increases, what happens to the value of sec? Eplain your reasoning. Section 9.7 Using Trigonometric Identities 7
6 33. MAKING AN ARGUMENT Your friend simplifies an epression and obtains sec tan sin. You simplify the same epression and obtain sin tan. Are your answers equivalent? Justify your answer. 3. HOW DO YOU SEE IT? The figure shows the unit circle and the angle. a. Is positive or negative?? tan? b. In what quadrant does the terminal side of lie? c. Is sin( ) positive or negative? cos( )? tan( )? (, y) y 37. DRAWING CONCLUSIONS Static friction is the amount of force necessary to keep a stationary object on a flat surface from moving. Suppose a book weighing W pounds is lying on a ramp inclined at an angle. The coefficient of static friction u for the book can be found using the equation uw = W. a. Solve the equation for u and simplify the result. b. Use the equation from part (a) to determine what happens to the value of u as the angle increases from 0 to PROBLEM SOLVING When light traveling in a medium (such as air) strikes the surface of a second medium (such as water) at an angle, the light begins to travel at a different angle. This change of direction is defined by Snell s law, n = n, where n and n are the indices of refraction for the two mediums. Snell s law can be derived from the equation 3. MODELING WITH MATHEMATICS A vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the Sun above the horizon is can be modeled by the equation below. Show that the equation below is equivalent to s = h cot. h s h sin(90 ) s = 36. THOUGHT PROVOKING Eplain how you can use a trigonometric identity to find all the values of for which sin = cos. n cot + = n cot + a. Simplify the equation to derive Snell s law.. air: n water: n b. What is the value of n when =, = 3, and n =? c. If =, then what must be true about the values of n and n? Eplain when this situation would occur. 39. WRITING Eplain how transformations of the graph of the parent function f () = sin support the cofunction identity sin ( π ) =. 0. USING STRUCTURE Verify each identity. a. ln sec = ln b. ln tan = ln ln Maintaining Mathematical Proficiency Find the value of for the right triangle. (Section 9.) Reviewing what you learned in previous grades and lessons Chapter 9 Trigonometric Ratios and Functions
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