Using Trigonometric Identities

9.7Using Trigonometric IdentitiesEssential QuestionHow can you verify a trigonometricidentity?Writing a Trigonometric IdentityWork with a partner. In the figure, the point(x, y) is on a circle of radius c with center atthe origin.y(x, y)a. Write an equation that relates a, b, and c.b. Write expressions for the sine and cosineratios of angle θ.cc. Use the results from parts (a) and (b) tofind the sum of sin2θ and cos2θ. What doyou observe?θbaxd. Complete the table to verify that the identity you wrote in part (c) is validfor angles (of your choice) in each of the four quadrants.θsin2 θcos2 θsin2 θ cos2 θQIQIIQIIIQIVWriting Other Trigonometric IdentitiesREASONINGABSTRACTLYTo be proficient in math,you need to know andflexibly use differentproperties of operationsand objects.Work with a partner. The trigonometric identity you derived in Exploration 1 iscalled a Pythagorean identity. There are two other Pythagorean identities. To derivethem, recall the four relationships:sin θtan θ —cos θcos θcot θ —sin θ1sec θ —cos θ1csc θ —sin θa. Divide each side of the Pythagorean identity you derived in Exploration 1by cos2θ and simplify. What do you observe?b. Divide each side of the Pythagorean identity you derived in Exploration 1by sin2θ and simplify. What do you observe?Communicate Your Answer3. How can you verify a trigonometric identity?4. Is sin θ cos θ a trigonometric identity? Explain your reasoning.5. Give some examples of trigonometric identities that are different than those inExplorations 1 and 2.Section 9.7hsnb alg2 pe 0907.indd 513Using Trigonometric Identities5132/5/15 1:53 PM

9.7LessonWhat You Will LearnUse trigonometric identities to evaluate trigonometric functions andsimplify trigonometric expressions.Core VocabulVocabularylarryVerify trigonometric identities.trigonometric identity, p. 514Previousunit circleUsing Trigonometric IdentitiesRecall that when an angle θ is in standardposition with its terminal side intersectingthe unit circle at (x, y), then x cos θ andy sin θ. Because (x, y) is on a circlecentered at the origin with radius 1, itfollows thatSTUDY TIPNote that sin2 θ represents(sin θ)2 and cos2 θrepresents (cos θ)2.yr 1(cos θ, sin θ) (x, y)θxx2 y2 1andcos2 θ sin2 θ 1.The equation cos2 θ sin2 θ 1 is true for any value of θ. A trigonometric equationthat is true for all values of the variable for which both sides of the equation aredefined is called a trigonometric identity. In Section 9.1, you used reciprocalidentities to find the values of the cosecant, secant, and cotangent functions. Theseand other fundamental trigonometric identities are listed below.Core ConceptFundamental Trigonometric IdentitiesReciprocal Identities1csc θ —sin θ1sec θ —cos θ1cot θ —tan θTangent and Cotangent Identitiessin θtan θ —cos θcos θcot θ —sin θPythagorean Identitiessin2 θ cos2 θ 11 tan2 θ sec2 θ1 cot2 θ csc2 θπcos — θ sin θ2πtan — θ cot θ2cos( θ) cos θtan( θ) tan θCofunction Identitiesπsin — θ cos θ2()()()Negative Angle Identitiessin( θ) sin θIn this section, you will use trigonometric identities to do the following. Evaluate trigonometric functions. Simplify trigonometric expressions. Verify other trigonometric identities.514Chapter 9hsnb alg2 pe 0907.indd 514Trigonometric Ratios and Functions2/5/15 1:54 PM

Finding Trigonometric Values4πGiven that sin θ — and — θ π, find the values of the other five trigonometric52functions of θ.SOLUTIONStep 1 Find cos θ.sin2 θ cos2 θ 1Write Pythagorean identity.2( 45 ) cos θ 14Substitute — for sin θ.52—4 242cos2 θ 1 —Subtract — from each side.559cos2 θ —Simplify.253cos θ —Take square root of each side.53cos θ —Because θ is in Quadrant II, cos θ is negative.5Step 2 Find the values of the other four trigonometric functions of θ using the valuesof sin θ and cos θ.43 ——5543sin θcos θcot θ — — —tan θ — — —cos θ3sin θ434 ——55()()151csc θ — — —sin θ44—5151sec θ — — —cos θ33 —5Simplifying Trigonometric ExpressionsπSimplify (a) tan — θ sin θ and (b) sec θ tan2 θ sec θ.2()SOLUTIONπa. tan — θ sin θ cot θ sin θ2cos θ — (sin θ)sin θ cos θ()Cofunction identity( )Cotangent identitySimplify.b. sec θ tan2 θ sec θ sec θ(sec2 θ 1) sec θPythagorean identity sec3 θ sec θ sec θDistributive Property sec3 θSimplify.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com16trigonometric functions of θ.π21. Given that cos θ — and 0 θ —, find the values of the other fiveSimplify the expression.2. sin x cot x sec x3. cos θ cos θ sin2 θSection 9.7hsnb alg2 pe 0907.indd 515tan x csc xsec x4. —Using Trigonometric Identities5152/5/15 1:54 PM

Verifying Trigonometric IdentitiesYou can use the fundamental identities from this chapter to verify new trigonometricidentities. When verifying an identity, begin with the expression on one side. Usealgebra and trigonometric properties to manipulate the expression until it is identicalto the other side.Verifying a Trigonometric Identitysec2 θ 1Verify the identity — sin2 θ.sec2 θSOLUTIONsec2 θ 1sec θsec2 θsec θ1sec θ — ——222Write as separate fractions.2( )1 1 —sec θSimplify. 1 cos2 θReciprocal identity sin2 θPythagorean identityNotice that verifying an identity is not the same as solving an equation. Whenverifying an identity, you cannot assume that the two sides of the equation are equalbecause you are trying to verify that they are equal. So, you cannot use any propertiesof equality, such as adding the same quantity to each side of the equation.Verifying a Trigonometric Identitycos xVerify the identity sec x tan x —.1 sin xSOLUTIONLOOKING FORSTRUCTURETo verify the identity, youmust introduce 1 sin xinto the denominator.Multiply the numeratorand the denominator by1 sin x so you get anequivalent expression.1sec x tan x — tan xcos xReciprocal identitysin x1 — —cos x cos xTangent identity1 sin x —cos xAdd fractions.1 sin x 1 sin x — —cos x1 sin x1 sin xMultiply by —.1 sin x1 sin2 x ——cos x(1 sin x)Simplify numerator.cos2 x ——cos x(1 sin x)Pythagorean identitycos x —1 sin xSimplify. Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comVerify the identity.516Chapter 9hsnb alg2 pe 0907.indd 5165. cot( θ) cot θ6. csc2 x(1 sin2 x) cot2 x7. cos x csc x tan x 18. (tan2 x 1)(cos2 x 1) tan2 xTrigonometric Ratios and Functions2/5/15 1:54 PM

9.7ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING Describe the difference between a trigonometric identity and a trigonometric equation.2. WRITING Explain how to use trigonometric identities to determine whether sec( θ) sec θ orsec( θ) sec θ.Monitoring Progress and Modeling with MathematicsIn Exercises 3 –10, find the values of the other fivetrigonometric functions of θ. (See Example 1.)π2133. sin θ —, 0 θ —ERROR ANALYSIS In Exercises 21 and 22, describe andcorrect the error in simplifying the expression.21. 22. 3π27104. sin θ —, π θ —3 π7 25. tan θ —, — θ π2 π5 26. cot θ —, — θ π3π2561 sin2 θ 1 (1 cos2 θ ) 1 1 cos2 θ cos2 θcos x 1tan x csc x — —sin x sin xcos x —sin2 x 7. cos θ —, π θ —9 3π4 28. sec θ —, — θ 2πIn Exercises 23–30, verify the identity. (See Examples 3and 4.)3π29. cot θ 3, — θ 2π23. sin x csc x 13π25324. tan θ csc θ cos θ 1( π2 )πsin( x ) tan x sin x2πcos( θ ) 12 1 28.10. csc θ —, π θ —25. cos — x cot x cos xIn Exercises 11–20, simplify the expression.(See Example 2.)26.12. cos θ (1 tan2 θ)11. sin x cot xsin( θ)cos( θ)cos2 xcot x13. —14. —2πcos — x215. —csc x()( π2 )16. sin — θ sec θcsc2 x cot2 xsin( x) cot xcos2 x tan2( x) 1cos x17. ——18. ——2πcos — θ219. — cos2 θcsc θπsec x sin x cos — x220. ———1 sec x()()——27. ——1 sin( θ)1 cos xsin xsin x1 cos x29. — — 2 csc xsin x1 cos( x)30. —— csc x cot x31. USING STRUCTURE A function f is odd whenf ( x) f(x). A function f is even whenf ( x) f (x). Which of the six trigonometricfunctions are odd? Which are even? Justify youranswers using identities and graphs.32. ANALYZING RELATIONSHIPS As the value of cos θincreases, what happens to the value of sec θ? Explainyour reasoning.Section 9.7hsnb alg2 pe 0907.indd 517sin2( x)tan x cos2 x—2Using Trigonometric Identities5172/5/15 1:54 PM

33. MAKING AN ARGUMENT Your friend simplifies37. DRAWING CONCLUSIONS Static friction is the amountan expression and obtains sec x tan x sin x. Yousimplify the same expression and obtain sin x tan2 x.Are your answers equivalent? Justify your answer.of force necessary to keep a stationary object on aflat surface from moving. Suppose a book weighingW pounds is lying on a ramp inclined at an angle θ.The coefficient of static friction u for the book can befound using the equation uW cos θ W sin θ.34. HOW DO YOU SEE IT? The figure shows the unitcircle and the angle θ.a. Solve the equation for u and simplify the result.a. Is sin θ positive or negative? cos θ? tan θ?b. Use the equation from part (a) to determine whathappens to the value of u as the angle θ increasesfrom 0 to 90 .b. In what quadrant does the terminal side of θ lie?c. Is sin( θ) positive or negative? cos( θ)?tan( θ)?38. PROBLEM SOLVING When light traveling in a medium(such as air) strikes the surface of a second medium(such as water) at an angle θ1, the light begins totravel at a different angle θ2. This change of directionis defined by Snell’s law, n1 sin θ1 n2 sin θ2, wheren1 and n2 are the indices of refraction for the twomediums. Snell’s law can be derived from the equationy(x, y)θxn1n2 cot2 θ1 1 cot2 θ2 1——— ———.35. MODELING WITH MATHEMATICS A vertical gnomon(the part of a sundial that projects a shadow) hasheight h. The length s of the shadow cast by thegnomon when the angle of the Sun above the horizonis θ can be modeled by the equation below. Show thatthe equation below is equivalent to s h cot θ.h sin(90 θ)s ——sin θhsair: n1θ1water: n2θ2a. Simplify the equation to derive Snell’s law.b. What is the value of n1 when θ1 55 , θ2 35 ,and n2 2?c. If θ1 θ2, then what must be true about thevalues of n1 and n2? Explain when this situationwould occur.39. WRITING Explain how transformations of thegraph of the parent function f (x) sin x support theπcofunction identity sin — θ cos θ.2θ)(40. USING STRUCTURE Verify each identity.a. ln sec θ ln cos θ 36. THOUGHT PROVOKING Explain how you can use atrigonometric identity to find all the values of x forwhich sin x cos x.Maintaining Mathematical Proficiencyb. ln tan θ ln sin θ ln cos θ Reviewing what you learned in previous grades and lessonsFind the value of x for the right triangle. (Section 9.1)41.42.43.1311x730 45 x518Chapter 9hsnb alg2 pe 0907.indd 51860 xTrigonometric Ratios and Functions2/5/15 1:54 PM

defi ned is called a trigonometric identity. In Section 9.1, you used reciprocal identities to fi nd the values of the cosecant, secant, and cotangent functions. These and other fundamental trigonometric identities are listed below. STUDY TIP Note that sin2 θ represents (sin θ)2 and cos2 θ represents (cos θ)2. trigonometric identity, p. 514 .