SECTION 6.2 Trigonometric Functions: Unit Circle Approach 397 OBJECTIVES 1 Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle (p. 398) 2 Find the Exact Values of the Trigonometric Functions of Quadrantal Angles (p. 399) 3 Find the Exact Values of the Trigonometric Functions of 4 45 π = ° (p. 401) 4 Find the Exact Values of the Trigonometric Functions of 6 30 π = ° and 3 60 π = ° (p. 402) 5 Find the Exact Values of the Trigonometric Functions for Integer Multiples of 6 30 , 4 45 , π π = ° = ° and 3 60 π = ° (p. 404) 6 Use a Calculator to Approximate the Value of a Trigonometric Function (p. 405) 7 Use a Circle of Radius r to Evaluate the Trigonometric Functions (p. 406) 6.2 Trigonometric Functions: Unit Circle Approach Now Work the ‘Are You Prepared?’ problems on page 408. • Geometry Essentials (Appendix A, Section A.2, pp. A14–A19) • Unit Circle (Section 1.6, p. 49) • Symmetry (Section 1.3, pp. 21–23) • Functions (Section 2.1, pp. 61–73) PREPARING FOR THIS SECTION Before getting started, review the following: We now introduce the trigonometric functions using the unit circle. The Unit Circle Recall that the unit circle is a circle whose radius is 1 and whose center is at the origin of a rectangular coordinate system. Also recall that any circle of radius r has circumference of length r 2 . π Therefore, the unit circle radius 1 ( ) = has a circumference of length 2 .π In other words, for 1 revolution around the unit circle the length of the arc is 2π units. The following discussion sets the stage for defining the trigonometric functions using the unit circle. Let t be any real number. Position the t -axis so that it is vertical with the positive direction up. Place this t -axis in the xy -plane so that t 0 = is located at the point 1, 0 ( ) in the xy -plane. If t 0, ≥ let s be the distance from the origin to t on the t -axis. See the red portion of Figure 20(a). Beginning at the point 1, 0 ( ) on the unit circle, travel s t = units in the counterclockwise direction along the circle, to arrive at the point P x y , . ( ) = In this sense, the length s t = units is being wrapped around the unit circle. If t 0, < we begin at the point 1, 0 ( ) on the unit circle and travel s t = units in the clockwise direction to arrive at the point P x y , . ( ) = See Figure 20(b). If t 2π > or if t 2 ,π <− it will be necessary to travel around the unit circle more than once before arriving at the point P. Do you see why? Let’s describe this process another way. Picture a string of length s t = units being wrapped around a circle of radius 1 unit. Start wrapping the string around the circle at the point 1, 0 . ( ) If t 0, ≥ wrap the string in the counterclockwise direction; if t 0, < wrap the string in the clockwise direction.The point P x y , ( ) = is the point where the string ends. This discussion tells us that, for any real number t, we can locate a unique point P x y , ( ) = on the unit circle. We call P the point on the unit circle that corresponds to t. This is the important idea here. No matter what real number t is chosen, there is a unique point P on the unit circle corresponding to it. The coordinates of the point P x y , ( ) = on the unit circle corresponding to the real number t are used to define the six trigonometric functions of t. Figure 20 y x t 0 s 5 t units 21 1 21 21 1 21 (1, 0) s 5 t units P 5 (x, y) (a) y x t 0 s 5|t | units (1, 0) s 5 |t | units P 5 (x, y) (b)
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