Solved: For individual or collaborative investigation (Exercises 7782) (This discussion

Chapter 5, Problem 78

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For individual or collaborative investigation (Exercises 7782) (This discussion applies to functions of both angles and real numbers.) The result of Example 3 in this section can be written as an identity. cos1180 U2 = cos U This 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 of the form 1Q t U2, 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. e 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 is positive in that quadrant, use a + sign on the reduced form. If is negative in that quadrant, use a - sign. The reduced form will have that sign, as the function, and u as the argument. Example: Cosine is negative in quadrant II. Terminates on the x-axis (1+)+1* This is in quadrant II for small u. Same function cos1180 - u2 = -cos u 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 is positive in that quadrant, use a + sign on the reduced form. If is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of as the function, and u as the argument. Example: Terminates on the y-axis Cosine is positive in quadrant IV. cos1270 + u2 = + sin u (or sin u, as it is usually written) (1+)+1* This is in quadrant IV for small u. Cofunctions Use these ideas to write a reduction formula for each of the following cos1270 - u2