SECTION 6.2 Trigonometric Functions: Unit Circle Approach 399 Figure 22 y x u s 5 |t| (1, 0) P 5 (x, y) (b) y x u 5 t radians u 5 t radians s 5 t units, t $ 0 s 5 |t| units, t , 0 (1, 0) P 5 (x, y) P 5 (x, y) (c) y x (1, 0) (d) y x t 21 (1, 0) 1 P 5 (x, y) 21 21 1 21 21 1 21 21 1 21 (a) Trigonometric Functions of Angles Let P x y , ( ) = be the point on the unit circle corresponding to the real number t. See Figure 22(a). Let θ be the angle in standard position, measured in radians, whose terminal side is the ray from the origin through P and whose arc length is t . See Figure 22(b). Since the unit circle has radius 1 unit, if s t = units, then from the arc length formula s r , θ = we have t θ = radians. See Figures 22(c) and (d). The point P x y , ( ) = on the unit circle that corresponds to the real number t is also the point P on the terminal side of the angle t θ = radians. As a result, we can say that t sin sinθ = ↑ ↑ Real number t radians θ = and so on. We can now define the trigonometric functions of the angle .θ Even though the trigonometric functions can be viewed both as functions of real numbers and as functions of angles, it is customary to refer to trigonometric functions of real numbers and trigonometric functions of angles collectively as the trigonometric functions . We shall follow this practice from now on. If an angle θ is measured in degrees, we shall use the degree symbol when writing a trigonometric function of ,θ as, for example, in ° sin30 and ° tan45 . If an angle θ is measured in radians, then no symbol is used when writing a trigonometric function of ,θ as, for example, in cosπ and sec 3 . π Finally, since the values of the trigonometric functions of an angle θ are determined by the coordinates of the point P x y , ( ) = on the unit circle corresponding to ,θ the units used to measure the angle θ are irrelevant. For example, it does not matter whether we write 2 θ π = radians or 90 . θ = ° The point on the unit circle corresponding to this angle is P 0, 1 . ( ) = As a result, sin 2 sin90 1 and cos 2 cos90 0 π π = ° = = ° = 2 Find the Exact Values of the Trigonometric Functions of Quadrantal Angles To find the exact value of a trigonometric function of an angle θ or a real number t requires that we locate the point P x y , ( ) = on the unit circle that corresponds to t. This is not always easy to do. In the examples that follow, we will evaluate the trigonometric functions of certain angles or real numbers for which this process is relatively easy. A calculator will be used to evaluate the trigonometric functions of most other angles. DEFINITION Trigonometric Functions of an Angle θ If t θ = radians, the six trigonometric functions of the angle θ are defined as t t t t t t sin sin cos cos tan tan csc csc sec sec cot cot θ θ θ θ θ θ = = = = = =
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