698 CHAPTER 7 Trigonometric Identities and Equations In the same way, S has coordinates 1cos A, sin A2, and R has coordinates 1cos1A - B2, sin1A - B22. Angle SOQ also equals A - B. The central angles SOQ and POR are equal, so chords PR and SQ are equal. Because PR = SQ, by the distance formula, 2 3cos1A - B2 - 142 + 3sin1A - B2 - 042 d = 21x2 - x122 + 1y 2 - y122 = 21cos A - cos B22 + 1sin A - sin B22. Square each side of this equation. Then square each expression, remembering that for any values of x and y, 1x - y22 = x2 - 2xy + y2. 3cos1A - B2 - 142 + 3sin1A - B2 - 042 = 1cos A - cos B22 + 1sin A - sin B22 cos21A - B2 - 2 cos1A - B2 + 1 + sin21A - B2 = cos2 A - 2 cos A cos B + cos2 B + sin2 A - 2 sin A sin B + sin2 B For any value of x, sin2 x + cos2 x = 1, so we can rewrite the equation. 2 - 2 cos1A - B2 = 2 - 2 cos A cos B - 2 sin A sin B Use sin2 x + cos2 x = 1 three times and add like terms. -2 cos1A - B2 = -2 cos A cos B - 2 sin A sin B Subtract 2. cos1A −B2 =cos A cos B +sin A sin B Divide by -2. This is the identity for cos1A - B2. Although Figure 4 on the previous page shows angles A and B in the second and first quadrants, respectively, this result is the same for any values of these angles. To find a similar expression for cos1A + B2, rewrite A + B as A - 1-B2 and use the identity for cos1A - B2. cos1A + B2 = cos3A - 1-B24 Definition of subtraction = cos A cos1-B2 + sin A sin1-B2 Cosine difference identity = cos A cos B + sin A1-sin B2 Even-odd identities cos1A +B2 =cos A cos B −sin A sin B Multiply. Cosine of a Sum or Difference cos1A +B2 =cos A cos B −sin A sin B cos1A −B2 =cos A cos B +sin A sin B These identities are important in calculus and useful in certain applications. For example, the method shown in Example 1 can be applied to find an exact value for cos 15°.
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