CC-32 Common Core State Standards MACC.92.F-TF.3.8 Prove the Pythagorean identity sin2(x) + cos2(x) and se it to find sin(x), cos(x), or tan(x), given sin(x), cos(x), or tan(x), and the qadrant of the angle. MP, MP 2, MP 3, MP 4 Objective To verify trigonometric identities y y Grahs of rational fnctions had holes like these. O O What cold be the fnction for each grah? Exlain yor reasoning. MATHEMATICAL PRACTICES Yo may recognize x 2 5x - 6 as an eqation that yo are to solve to find the few, if any, vales of x that make the eqation tre. On the other hand, yo may recognize x5 x 2, as an identity, an eqation that is tre for all vales of x for which the x3 5 exressions in the eqation are defined. (Here, x 3 is not defined for x 0.) x Lesson Vocablary trigonometric identity A trigonometric identity in one variable is a trigonometric eqation that is tre for all vales of the variable for which all exressions in the eqation are defined. Essential Understanding The interrelationshis among the six basic trigonometric fnctions make it ossible to write trigonometric exressions in varios eqivalent forms, some of which can be significantly easier to work with than others in mathematical alications. Some trigonometric identities are definitions or follow immediately from definitions. Key Concet Recirocal Identities Basic Identities csc sin sec cos tan cot sin csc cos sec cot tan sin tan cos Cotangent Identity cos cot sin The domain of validity of an identity is the set of vales of the variable for which all exressions in the eqation are defined. 60 60 Chater 4 and Eqations HSM5_A2Hon_SE_CC_32_TrKit.indd 60 Common Core HSM5_A2 8
Problem Finding the Domain of Validity What is the domain of validity of each trigonometric identity? How can an exression be ndefined? An exression cold contain a denominator that cold be zero or it cold contain an exression that is itself ndefined for some vales. A cos U sec U. The domain of cos is all real nmbers. The domain of sec excldes all zeros of sec (of which there are none) and all vales for which sec is ndefined (odd mltiles of 2 ). Therefore the domain of validity of cos sec is the set of real nmbers excet for the odd mltiles of 2. y 2 O sec cos 2 B sec U cos U. The domain of validity is the same as art (a), becase sec is not defined for odd mltiles of 2, and the odd mltiles of 2 are the zeros of cos. Got It?. What is the domain of validity of the trigonometric identity sin csc? Yo can se known identities to verify other identities. To verify an identity, yo can se reviosly known identities to transform one side of the eqation to look like the other side. Problem 2 Verifying an Identity Using Basic Identities Verify the identity. What is the domain of validity? What identity do yo know that yo can se? Look for a way to write the exression on the left in terms of sin and cos. The identity sec cos does the job. 2Hon_SE_CC_32_TrKit.indd 6 8/5/3 7:2 PM A (sin U)(sec U) tan U (sin )(sec ) sin # cos sin Recirocal Identity cos Simlify. tan The domain of sin is all real nmbers. The domains of sec and tan exclde all zeros of cos. These are the odd mltiles of 2. The domain of validity is the set of real nmbers excet for the odd mltiles of 2. B tan U cot U cot tan tan Definition of cotangent Simlify. The domain of cot excldes mltiles of. Also, cot 0 at the odd mltiles of 2. The domain of validity is the set of real nmbers excet all mltiles of 2. csc Got It? 2. Verify the identity sec cot. What is the domain of validity? Lesson 4- CC-32 6 8/5/3 6 7:2 PM
Yo can se the nit circle and the Pythagorean Theorem to verify another identity. The circle with its center at the origin with a radis of is called the nit circle, and has an eqation x 2 + y 2. (cos, sin ) y sin x cos Every angle determines a niqe oint on the nit circle with x- and y-coordinates (x, y) (cos, sin ). This form allows yo to Therefore, for every angle, (cos )2 + (sin )2 or cos2 + sin2. hsm2_a2_se_c4_l0_t000.ai write the identity withot sing arentheses. This is a Pythagorean identity. Yo will verify two others in Problem 3. Yo can se the basic and Pythagorean identities to verify other identities. To rove identities, transform the exression on one side of the eqation to the exression on the other side. It often hels to write everything in terms of sines and cosines. Problem 3 Verifying a Pythagorean Identity With which side shold yo work? It sally is easier to begin with the more comlicated-looking side. Verify the Pythagorean identity + tan2 U sec2 U. ( sin )2 + tan2 + cos + sin2 cos2 Simlify. cos2 sin2 + cos2 cos2 Find a common denominator. cos2 + sin2 cos2 Add. cos2 Pythagorean identity sec2 Recirocal identity Yo have transformed the exression on the left side of the eqation to become the exression on the right side. The eqation is an identity. Got It? 3. a. Verify the third Pythagorean identity, + cot2 csc2. b. Reasoning Exlain why the domain of validity is not the same for all three Pythagorean identities. Yo have now seen all three Pythagorean identities. Key Concet cos2 + sin2 62 62 Pythagorean Identities + tan2 sec2 + cot2 csc2 Chater 4 and Eqations HSM5_A2Hon_SE_CC_32_TrKit.indd 62 Common Core HSM5_A2 8
There are many trigonometric identities. Most do not have secific names. Problem 4 Verifying an Identity Verify the identity tan2 U sin2 U tan2 U sin2 U. How do yo begin when both sides look comlicated? It often is easier to collase a difference (or sm) into a rodct than to exand a rodct into a difference. 2Hon_SE_CC_32_TrKit.indd 63 8/5/3 7:2 PM sin2 - sin2 cos2 Tangent identity sin2 sin2 cos2 cos2 cos2 Use a common denominator. sin2 - sin2 cos2 cos2 Simlify. sin2 ( - cos2 ) cos2 Factor. sin2 (sin2 ) cos2 Pythagorean identity sin2 sin2 cos2 Rewrite the fraction. tan2 - sin2 tan2 sin2 Got It? 4. Verify the identity sec2 - sec2 cos2 tan2. Yo can se trigonometric identities to simlify trigonometric exressions. Problem 5 Simlifying an Exression What is a simlified trigonometric exression for csc U tan U? Write the exression. Then relace csc with sin. csc tan sin # tan sin sin # cos sin sin cos sin Relace tan with cos. Simlify. cos cos sec. sec Got It? 5. What is a simlified trigonometric exression for sec cot? Lesson 4- CC-32 63 8/5/3 63 7:2 PM
Lesson Check Do yo know HOW? Verify each identity.. tan csc sec 2. csc 2 - cot 2 3. sin tan sec - cos 4. Simlify tan cot - sin 2. Do yo UNDERSTAND? MATHEMATICAL PRACTICES 5. Vocablary How does the identity cos 2 + sin 2 relate to the Pythagorean Theorem? 6. Error Analysis A stdent simlified the exression 2 - cos 2 to - sin 2. What error did the stdent make? What is the correct simlified exression? Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice Verify each identity. Give the domain of validity for each identity. See Problems 4. 7. cos cot sin - sin 8. sin cot cos 9. cos tan sin 0. sin sec tan. cos sec 2. tan cot 3. sin csc 4. cot csc cos 5. csc - sin cot cos B Aly Simlify each trigonometric exression. 6. tan cot 7. - cos 2 8. sec 2-9. - csc 2 20. sec 2 cot 2 2. cos tan 22. sin cot 23. sin csc 24. sec cos sin 25. sin sec cot 26. sec 2 - tan 2 sin 27. cos tan tan 28. Think Abot a Plan Simlify the exression sec - cos. Can yo write everything in terms of sin, cos, or both? Are there any trigonometric identities that can hel yo simlify the exression? Simlify each trigonometric exression. 29. cos + sin tan 30. csc cos tan 3. tan (cot + tan ) 32. sin 2 + cos 2 + tan 2 33. sin ( + cot 2 ) 34. sin 2 csc sec 35. sec cos - cos 2 36. csc - cos cot 37. csc 2 ( - cos 2 csc ) 38. sin + cos cot 39. cos csc cot 40. sin2 csc sec tan See Problem 5. 64 Common Core
Exress the first trigonometric fnction in terms of the second. 4. sin, cos 42. tan, cos 43. cot, sin 44. csc, cot 45. cot, csc 46. sec, tan Verify each identity. 47. sin 2 tan 2 tan 2 - sin 2 48. sec - sin tan cos 49. sin cos (tan + cot ) 50. - sin cos cos + sin 5. sec cot + tan sin 52. cot + 22 csc 2 + 2 cot C Challenge 53. Exress cos csc cot in terms of sin. cos 54. Exress sec + tan in terms of sin. Use the identity sin 2 U + cos 2 U and the basic identities to answer the following qestions. Show all yor work. 55. Given that sin 0.5 and is in the first qadrant, what are cos and tan? 56. Given that sin 0.5 and is in the second qadrant, what are cos and tan? 57. Given that cos -0.6 and is in the third qadrant, what are sin and tan? 58. Given that sin 0.48 and is in the second qadrant, what are cos and tan? 59. Given that tan.2 and is in the first qadrant, what are sin and cos? 60. Given that tan 3.6 and is in the third qadrant, what are sin and cos? 6. Given that sin 0.2 and tan 6 0, what is cos? 62. The nit circle is a sefl tool for verifying identities. Use the diagram at the right to verify the identity sin( + ) -sin. a. Exlain why the y-coordinate of oint P is y sin( + ). b. Prove that the two triangles shown are congrent. (cos U, sin U) c. Use art (b) to show that the two ble segments are congrent. x d. Use art (c) to show that the y-coordinate of P is -sin. e. Use arts (a) and (d) to conclde that P sin( + ) -sin. Use the diagram in Exercise 62 to verify each identity. 63. cos( + ) -cos 64. tan( + ) tan Simlify each trigonometric exression. 65. cot2 - csc 2 tan 2 - sec 2 66. ( - sin )( + sin )csc 2 + CC-32 65
STEM 67. Physics When a ray of light asses from one medim into a second, the angle of incidence and the angle of refraction 2 are related by Snell s law: n sin n 2 sin 2, where n is the index of refraction of the first medim and n 2 is the index of refraction of the second medim. How are and 2 related if n 2 7 n? If n 2 6 n? If n 2 n? 2 66 Common Core