Common Core State Standards CC-32 Trigonometric

CC-32 Trigonometric IdentitiesCommon Core State StandardsMACC.912.F-TF.3.8 Prove the Pythagorean identitysin2(x) cos2(x) 1 and use it to find sin(x), cos(x), ortan(x), given sin(x), cos(x), or tan(x), and the quadrantof the angle.MP 1, MP 2, MP 3, MP 4ObjectiveTo verify trigonometric identitiesyypGraphs of rationalfunctions had holeslike these.OppuOpu11What could be the function for each graph? Explain your reasoning.MATHEMATICALPRACTICESYou may recognize x 2 5x - 6 as an equation that you are to solve to find the few,if any, values of x that make the equation true. On the other hand, you may recognizex5 x 2 , as an identity, an equation that is true for all values of x for which thex35expressions in the equation are defined. (Here, x 3 is not defined for x 0.)xLessonVocabulary trigonometricidentityA trigonometric identity in one variable is a trigonometric equation that is true for allvalues of the variable for which all expressions in the equation are defined.Essential Understanding The interrelationships among the six basictrigonometric functions make it possible to write trigonometric expressions in variousequivalent forms, some of which can be significantly easier to work with than others inmathematical applications.Some trigonometric identities are definitions or follow immediately from definitions.Key ConceptReciprocal IdentitiesTangent IdentityBasic Identitiescsc u sin1 u1sec u cosu1tan u cotu1sin u cscu1cos u secu1cot u tanusin utan u cos uCotangent Identitycos ucot u sin uThe domain of validity of an identity is the set of values of the variable for which allexpressions in the equation are defined.160160Chapter 14 Trigonometric Identities and EquationsHSM15 A2Hon SE CC 32 TrKit.indd160Common CoreHSM15 A28

Problem 1 Finding the Domain of ValidityWhat is the domain of validity of each trigonometric identity?How can anexpression beundefined?An expression couldcontain a denominatorthat could be zero orit could contain anexpression that is itselfundefined for somevalues.1A cos U secU.The domain of cos u is all real numbers. Thedomain of sec1 u excludes all zeros of sec u (ofwhich there are none) and all values u forwhich sec u is undefined (odd multiples of p2 ).Therefore the domain of validity ofcos u sec1 u is the set of real numbersexcept for the odd multiples of p2 .y1p2O1sec ucos up2pu1B sec U cosU.The domain of validity is the same as part (a), because sec u is not defined for oddmultiples of p2 , and the odd multiples of p2 are the zeros of cos u.Got It? 1. What is the domain of validity of the trigonometric identity sin u csc1 u ?You can use known identities to verify other identities. To verify an identity, you canuse previously known identities to transform one side of the equation to look like theother side.Problem 2 Verifying an Identity Using Basic IdentitiesVerify the identity. What is the domain of validity?What identity doyou know that youcan use?Look for a way to writethe expression on theleft in terms of sin uand cos u. The identitysec u cos1 u does thejob.2Hon SE CC 32 TrKit.indd1618/5/137:21 PMA (sin U)(sec U) tan U(sin u)(sec u) sin u# cos1 usin uReciprocal Identity cos uSimplify. tan uTangent IdentityThe domain of sin u is all real numbers. The domains of sec u and tan u exclude allzeros of cos u. These are the odd multiples of p2 . The domain of validity is the set ofreal numbers except for the odd multiples of p2 .B1 tan Ucot U11cot u 1tan u tan uDefinition of cotangentSimplify.The domain of cot u excludes multiples of p. Also, cot u 0 at the odd multiples of p2 .The domain of validity is the set of real numbers except all multiples of p2 .csc uGot It? 2. Verify the identity secu cot u. What is the domain of validity?Lesson 14-1CC-32Trigonometric IdentitiesTrigonometric Identities1618/5/131617:21 PM

You can use the unit circle and the Pythagorean Theorem to verifyanother identity. The circle with its center at the origin with aradius of 1 is called the unit circle, and has an equationx 2 y 2 1.(cos u, sin u)1y sin ux cos uEvery angle u determines a unique point on the unit circlewith x- and y-coordinates (x, y) (cos u, sin u).This form allows you toTherefore, for every angle u,(cos u)2 (sin u)2 1orcos2 u sin2 u 1.hsm12 a2 se c14 L01 t0001.aiwrite the identity withoutusing parentheses.This is a Pythagorean identity. You will verify two others in Problem 3.You can use the basic and Pythagorean identities to verify other identities. To proveidentities, transform the expression on one side of the equation to the expression on theother side. It often helps to write everything in terms of sines and cosines.Problem 3 Verifying a Pythagorean IdentityWith which sideshould you work?It usually is easier tobegin with the morecomplicated-looking side.Verify the Pythagorean identity 1 tan2 U sec2 U.( sin u )21 tan2 u 1 cos u 1 sin2 ucos2 uTangent IdentitySimplify. cos2 u sin2 u cos2 u cos2 uFind a common denominator. cos2 u sin2 ucos2 uAdd. 1cos2 uPythagorean identity sec2 uReciprocal identityYou have transformed the expression on the left side of the equation to become theexpression on the right side. The equation is an identity.Got It? 3. a. Verify the third Pythagorean identity, 1 cot2 u csc2 u.b. Reasoning Explain why the domain of validity is not the same for allthree Pythagorean identities.You have now seen all three Pythagorean identities.Key Conceptcos2 u sin2 u 1162162Pythagorean Identities1 tan2 u sec2 u1 cot2 u csc2 uChapter 14 Trigonometric Identities and EquationsHSM15 A2Hon SE CC 32 TrKit.indd162Common CoreHSM15 A28

There are many trigonometric identities. Most do not have specific names.Problem 4 Verifying an IdentityVerify the identity tan2 U sin2 U tan2 U sin2 U.How do you beginwhen both sides lookcomplicated?It often is easier tocollapse a difference (orsum) into a product thanto expand a product intoa difference.2Hon SE CC 32 TrKit.indd1638/5/137:21 PMsin2 u- sin2 ucos2 uTangent identity sin2 u sin2 u cos2 ucos2 ucos2 uUse a common denominator. sin2 u - sin2 u cos2 ucos2 uSimplify. sin2 u(1 - cos2 u)cos2 uFactor. sin2 u(sin2 u)cos2 uPythagorean identity sin2 usin2 ucos2 uRewrite the fraction.tan2 u - sin2 u tan2 u sin2 uTangent IdentityGot It? 4. Verify the identity sec2 u - sec2 u cos2 u tan2 u.You can use trigonometric identities to simplify trigonometric expressions.Problem 5 Simplifying an ExpressionWhat is a simplified trigonometric expression for csc U tan U?Write the expression.1Then replace csc u with sin u.csc u tan u 1sin u# tan u 1sin usin u# cosu sin usin u cos usin uReplace tan u with cos u.Simplify.1 cosu1cos u sec u. sec uGot It? 5. What is a simplified trigonometric expression for sec u cot u?Lesson 14-1CC-32Trigonometric IdentitiesTrigonometric Identities1638/5/131637:21 PM

Lesson CheckDo you know HOW?1. tan u csc u sec u2. csc2 u - cot2 u 16. Error Analysis A student simplified the expression2 - cos2 u to 1 - sin2 u. What error did the studentmake? What is the correct simplified expression?3. sin u tan u sec u - cos u4. Simplify tan u cot u - sin2 u.MATHEMATICALPractice and Problem-Solving ExercisesPracticePRACTICES5. Vocabulary How does the identitycos2 u sin2 u 1 relate to the PythagoreanTheorem?Verify each identity.AMATHEMATICALDo you UNDERSTAND?PRACTICESSee Problems 1–4.Verify each identity. Give the domain of validity for each identity.7. cos u cot u sin1 u - sin u8. sin u cot u cos u9. cos u tan u sin u10. sin u sec u tan u11. cos u sec u 112. tan u cot u 113. sin u csc u 114. cot u csc u cos u15. csc u - sin u cot u cos uSee Problem 5.Simplify each trigonometric expression.BApplycos2 usec2 u16. tan u cot u17. 1 -19. 1 - csc2 u20. sec2 u cot2 u21. cos u tan u22. sin u cot u23. sin u csc u24. sec u cos u sin u25. sin u sec u cot u26. sec2 u - tan2 u27. cos u tan u18.-1sin uu28. Think About a Plan Simplify the expression sec utan- cos u . Can you write everything in terms of sin u, cos u, or both? Are there any trigonometric identities that can help you simplify the expression?Simplify each trigonometric expression.29. cos u sin u tan u30. csc u cos u tan u31. tan u(cot u tan u)32. sin2 u cos2 u tan2 u33. sin u(1 cot2 u)34. sin2 u csc u sec u35. sec u cos u - cos2 u36. csc u - cos u cot u37. csc2 u(1 - cos2 u)38. sin u cos u cot u39.164164cos u csc ucot ucsc u40.sin2 u csc u sec utan uChapter 14 Trigonometric Identities and EquationsHSM15 A2Hon SE CC 32 TrKit.inddCommon164CoreHSM15 A28/5/

Express the first trigonometric function in terms of the second.41. sin u, cos u42. tan u, cos u43. cot u, sin u44. csc u, cot u45. cot u, csc u46. sec u, tan u48. sec u - sin u tan u cos u49. sin u cos u (tan u cot u) 1Verify each identity.47. sin2 u tan2 u tan2 u - sin2 u50.1 - sin ucos ucos u 1 sin usec u52. 1 cot u 1 2 2 csc2 u 2 cot u51. cot u tan u sin u53. Express cos u csc u cot u in terms of sin u.u54. Express sec ucos tan u in terms of sin u.Use the identity sin2 U cos2 U 1 and the basic identities to answer thefollowing questions. Show all your work.55. Given that sin u 0.5 and u is in the first quadrant, what are cos u and tan u?56. Given that sin u 0.5 and u is in the second quadrant, what are cos u and tan u?57. Given that cos u -0.6 and u is in the third quadrant, what are sin u and tan u?58. Given that sin u 0.48 and u is in the second quadrant, what are cos u and tan u?59. Given that tan u 1.2 and u is in the first quadrant, what are sin u and cos u?60. Given that tan u 3.6 and u is in the third quadrant, what are sin u and cos u?61. Given that sin u 0.2 and tan u 6 0, what is cos u?CChallenge2Hon SE CC 32 TrKit.indd165/13 7:21 PM62. The unit circle is a useful tool for verifying identities. Use the diagramat the right to verify the identity sin(u p) -sin u.a. Explain why the y-coordinate of point P issin(u p).b. Prove that the two triangles shown are congruent.c. Use part (b) to show that the two blue segments arecongruent.d. Use part (c) to show that the y-coordinate ofP is -sin u.e. Use parts (a) and (d) to conclude thatsin(u p) -sin u.(cos U, sin U)pu11xP1Use the diagram in Exercise 62 to verify each identity.63. cos(u p) -cos u1 y64. tan(u p) tan uSimplify each trigonometric expression.65.cot2 u - csc2 utan2 u - sec2 u66. (1 - sin u)(1 sin u)csc2 u 1Lesson 14-1CC-32Trigonometric IdentitiesTrigonometric Identities1658/5/131657:22 PM

STEM16616667. Physics When a ray of light passes from one medium into a second, the angle ofincidence u1 and the angle of refraction u2 are related by Snell’s law:n1 sin u1 n2 sin u2 , where n1 is the index of refraction of the first medium andn2 is the index of refraction of the second medium. How are u1 and u2 related ifn2 7 n1 ? If n2 6 n1 ? If n2 n1 ?u1u2Chapter 14 Trigonometric Identities and EquationsHSM15 A2Hon SE CC 32 TrKit.inddCommon166Core8/5/

Oct 18, 2015 · trigonometric functions make it possible to write trigonometric expressions in various equivalent forms, some of which can be significantly easier to work with than others in mathematical applications. Some trigonometric identities are definitions or follow immediately from definitions. Lesson Vocabulary trigonometric identity Lesson