700 CHAPTER 7 Trigonometric Identities and Equations Cofunction Identities For any angle u for which the functions are defined, the following hold true. cos190° −U2 =sin U cot190° −U2 =tan U sin190° −U2 =cos U sec190° −U2 =csc U tan190° −U2 =cot U csc190° −U2 =sec U The same identities can be obtained for a real number domain by replacing 90° with p 2. EXAMPLE 2 Using Cofunction Identities to Find U Find one value of u or x that satisfies each of the following. (a) cot u = tan 25° (b) sin u = cos1-30°2 (c) csc 3p 4 = sec x SOLUTION (a) Because tangent and cotangent are cofunctions, tan190° - u2 = cot u. cot u = tan 25° tan190° - u2 = tan 25° Cofunction identity 90° - u = 25° Set angle measures equal. u = 65° Solve for u. (b) sin u = cos1-30°2 cos190° - u2 = cos1-30°2 Cofunction identity 90° - u = -30° Set angle measures equal. u = 120° Solve for u. (c) csc 3p 4 = sec x csc 3p 4 = csca p 2 - xb Cofunction identity 3p 4 = p 2 - x Set angle measures equal. x = - p 4 Solve for x; p 2 - 3p 4 = 2p 4 - 3p 4 = - p 4 S Now Try Exercises 35 and 39. Sine and Tangent Sum and Difference Identities We can derive identities for sine by replacing u in sin u = cos190° - u2 with A + B. sin1A + B2 = cos390° - 1A + B24 Cofunction identity = cos3190° - A2 - B4 Distribute negative sign and regroup. = cos190° - A2 cos B + sin190° - A2 sin B sin1A +B2 =sin A cos B +cos A sin B Cofunction identities Cosine difference identity
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