760 CHAPTER 7 Trigonometric Identities and Equations Chapter 7 Test Prep New Symbols sin−1 x 1arcsin x2 inverse sine of x cos−1 x 1arccos x2 inverse cosine of x tan−1 x 1arctan x2 inverse tangent of x cot−1 x 1arccot x2 inverse cotangent of x sec−1 x 1arcsec x2 inverse secant of x csc−1 x 1arccsc x2 inverse cosecant of x Quick Review 7.1 Fundamental Identities Reciprocal Identities cot u = 1 tan u sec u = 1 cos u csc u = 1 sin u Quotient Identities tan u = sin u cos u cot u = cos u sin u Pythagorean Identities sin2 u + cos2 u = 1 tan2 u + 1 = sec2 u 1 + cot2 u = csc2 u Even-Odd Identities sin1-u2 = - sin u cos1-u2 = cos u tan1-u2 = - tan u csc1-u2 = - csc u sec1-u2 = sec u cot1-u2 = - cot u 7.2 Verifying Trigonometric Identities See the box titled Hints for Verifying Identities in Section 7.2. If u is in quadrant IV and sin u = - 3 5 , find csc u, cos u, and sin1-u2. csc u = 1 sin u = 1 - 3 5 = - 5 3 sin2 u + cos2 u = 1 Pythagorean identity a- 3 5b 2 + cos2 u = 1 Substitute. cos2 u = 16 25 A - 3 5B 2 = 9 25 ; Subtract 9 25 . cos u = +B16 25 cos u = 4 5 sin1-u2 = - sin u = - a- 3 5b = 3 5 Even-odd identity Reciprocal identity cos u is positive in quadrant IV. 7.3 Sum and Difference Identities Cofunction Identities cos190° - u2 = sin u cot190° - u2 = tan u sin190° - u2 = cos u sec190° - u2 = csc u tan190° - u2 = cot u csc190° - u2 = sec u Concepts Examples Find one value of u such that tan u = cot 78°. t an u = cot 78° cot190° - u2 = cot 78° Cofunction identity 90° - u = 78° Set angles equal. u = 12° Solve for u.
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