699 7.3 Sum and Difference Identities Cofunction Identities We can use the identity for the cosine of the difference of two angles and the fundamental identities to derive cofunction identities, presented previously for values of u in the interval 30°, 90°4. For example, substituting 90° for A and u for B in the identity for cos1A - B2 gives the following. cos190° −U2 = cos 90° cos u + sin 90° sin u Cosine difference identity = 0 # cos u + 1 # sin u cos 90° = 0 and sin 90° = 1 =sin U Simplify. This result is true for any value of u because the identity for cos1A - B2 is true for any values of A and B. EXAMPLE 1 Finding Exact Cosine Function Values Find the exact value of each expression. (a) cos 15° (b) cos 5p 12 (c) cos 87° cos 93° - sin 87° sin 93° SOLUTION (a) To find cos 15°, we write 15° as the sum or difference of two angles with known function values, such as 45° and 30°, because 15° = 45° - 30°. (We could also use 60° - 45°.) Then we use the cosine difference identity. cos 15° = cos145° - 30°2 15° = 45° - 30° = cos 45° cos 30° + sin 45° sin 30° Cosine difference identity = 22 2 # 23 2 + 22 2 # 1 2 Substitute known values. = 26 + 22 4 Multiply, and then add fractions. (b) cos 5p 12 = cos a p 6 + p 4b 5p 12 = 2p 12 + 3p 12 = p 6 + p 4 = cos p 6 cos p 4 - sin p 6 sin p 4 Cosine sum identity = 23 2 # 22 2 - 1 2 # 22 2 Substitute known values. = 26 - 22 4 Multiply, and then subtract fractions. (c) cos 87° cos 93° - sin 87° sin 93° = cos187° + 93°2 Cosine sum identity = cos 180° Add. = -1 cos 180° = -1 This screen supports the solution in Example 1(b) by showing that the decimal approximations for cos 5p 12 and 26 - 22 4 agree. S Now Try Exercises 11, 15, and 19.
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