710 CHAPTER 7 Trigonometric Identities and Equations Relating Concepts For individual or collaborative investigation (Exercises 107 – 112) (This discussion applies to functions of both angles and real numbers.) Consider the following. cos1180° - u2 = cos 180° cos u + sin 180° sin u Cosine difference identity = 1-12 cos u + 102 sin u cos 180° = -1 and sin 180° = 0 = - cos u Simplify. cos1180° −U2 = −cos U is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus u to a function of u alone. Another example of a reduction formula is cos1270° +U2 =sin U. Here is an interesting method for quickly determining a reduction formula for a trigonometric function ƒ of the form ƒ1QtU2, where Q is a quadrantal angle. There are two cases to consider, and in each case, think of U as a small positive angle in order to determine the quadrant in which Q{u will lie. Case 1 Q is a quadrantal angle whose terminal side lies along the x-axis. Determine the quadrant in which Q{u will lie for a small positive angle u. If the given function f is positive in that quadrant, use a + sign on the reduced form. If f is negative in that quadrant, use a - sign. The reduced form will have that sign, f as the function, and u as the argument. Example: Cosine is negative in quadrant II. Terminates on the x@axis cos1180° - u2 = -cos u (1+)+1* This is in quadrant II for small u. Same function cos1270° + u2 = + sin u 1or sin u, as it is usually written2 Terminates on the y@axis Cosine is positive in quadrant IV. (1+)+1* This is in quadrant IV for small u. Cofunctions Case 2 Q is a quadrantal angle whose terminal side lies along the y-axis. Determine the quadrant in which Q{u will lie for a small positive angle u. If the given function f is positive in that quadrant, use a + sign on the reduced form. If f is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of f as the function, and u as the argument. Example: Use these ideas to write a reduction formula for each of the following. 107. cos190° + u2 108. cos1270° - u2 109. cos1180° + u2 110. cos1270° + u2 111. sin1180° + u2 112. tan1270° - u2
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