402 CHAPTER 6 Trigonometric Functions Figure 26 30 60 90 °− °− ° triangle 608 308 a (a) b c 5 1 c 5 1 c 5 1 608 608 308 308 a a (b) b 1 608 308 a 5 b 5 1 2 (c) 3 2 Finding the Exact Value of a Trigonometric Expression Find the exact value of each expression. (a) sin45 cos180 ° ° (b) tan 4 sin 3 2 π π − (c) sec 4 csc 2 2 π π ( ) + EXAMPLE 5 Solution (a) ( ) ° ° = ⋅ − = − sin45 cos180 2 2 1 2 2 ↑ ↑ From Example 4 From Table 2 (b) tan 4 sin 3 2 1 1 2 π π ( ) − = − − = ↑ ↑ From Example 4 From Table 2 (c) sec 4 csc 2 2 1 2 1 3 2 2 π π ( ) ( ) + = + = + = Now Work PROBLEM 35 4 Find the Exact Values of the Trigonometric Functions of π = ° 6 30 and π = ° 3 60 Consider a right triangle in which one of the angles is 6 30 . π = ° It then follows that the third angle is 3 60 . π = ° Figure 26(a) illustrates such a triangle with hypotenuse of length 1. Our problem is to determine a and b. Begin by placing next to the triangle in Figure 26(a) another triangle congruent to the first, as shown in Figure 26(b). Notice that we now have a triangle whose three angles each equal 60 .° This triangle is therefore equilateral, so each side is of length 1. This means the base is a2 1, = and so a 1 2 . = By the Pythagorean Theorem, b satisfies the equation a b c , 2 2 2 + = so we have + = + = = − = = a b c b b b 1 4 1 1 1 4 3 4 3 2 2 2 2 2 2 This results in Figure 26(c). = = a c 1 2 , 1 b 0 > because b is the length of the side of a triangle.
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