682 CHAPTER 7 Trigonometric Identities and Equations Fundamental Identities Recall that a function is even if ƒ1 −x2 =ƒ1x2 for all x in the domain of ƒ, and a function is odd if ƒ1 −x2 = −ƒ1x2 for all x in the domain of ƒ. We have used graphs to classify the trigonometric functions as even or odd. We can also use Figure 1 to do this. As suggested by the circle in Figure 1, an angle u having the point 1x, y2 on its terminal side has a corresponding angle -u with the point 1x, -y2 on its terminal side. From the definition of sine, we see that sin1-u2 and sin u are negatives of each other. That is, sin1-u2 = -y r and sin u = y r , so sin1−U2 = −sin U Sine is an odd function. This is an example of an identity, an equation that is satisfied by every value in the domain of its variable. Some examples from algebra follow. x2 - y2 = 1x + y21x - y2 x1x + y2 = x2 + xy Identities x2 + 2xy + y2 = 1x + y22 Figure 1 shows an angle u in quadrant II, but the same result holds for u in any quadrant. The figure also suggests the following identity for cosine. cos1-u2 = x r and cos u = x r cos1−U2 =cos U Cosine is an even function. We use the identities for sin1-u2 and cos1-u2 to find tan1-u2 in terms of tan u. tan1-u2 = sin1-u2 cos1-u2 = -sin u cos u = - sin u cos u tan1−U2 = −tan U Tangent is an odd function. The reciprocal identities are used to determine that cosecant and cotangent are odd functions and secant is an even function. These even-odd identities, together with the reciprocal, quotient, and Pythagorean identities, make up the fundamental identities. 7.1 Fundamental Identities ■ Fundamental Identities ■ Uses of the Fundamental Identities sin(–U) = – = –sin U y r O x y y x (x, y) –y r (x, –y) –U U r Figure 1 NOTE In trigonometric identities, u can represent an angle in degrees or radians, or a real number.
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