SECTION 8.1 Right Triangle Trigonometry; Applications 547 Finding the Value of Trigonometric Functions from a Right Triangle Find the exact value of the six trigonometric functions of the angleθ in Figure 3. Solution EXAMPLE 1 In Figure 3 the two given sides of the triangle are c a Hypotenuse 5 Adjacent 3 = = = = To find the length of the opposite side, use the Pythagorean Theorem. Adjacent Opposite Hypotenuse 3 Opposite 5 Opposite 25 9 16 Opposite 4 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) + = + = = − = = Now that the lengths of the three sides are known, use the ratios in (1) to find the value of each of the six trigonometric functions. sin Opposite Hypotenuse 4 5 cos Adjacent Hypotenuse 3 5 tan Opposite Adjacent 4 3 csc Hypotenuse Opposite 5 4 sec Hypotenuse Adjacent 5 3 cot Adjacent Opposite 3 4 θ θ θ θ θ θ = = = = = = = = = = = = Figure 3 5 Opposite 3 u Now Work PROBLEM 9 The values of the trigonometric functions of an acute angle are ratios of the lengths of the sides of a right triangle. This way of viewing the trigonometric functions leads to many applications and, in fact, was the point of view used by early mathematicians (before calculus) in studying the subject of trigonometry. 4 b a (c) Figure 4 4 in. 4 in. 4 in. 4 in. 12 in. 4 in. 4 in. b (b) (a) b a Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. See Figure 4(a). After marking off a length of 4 inches from each edge, the sides are bent up at an angle .θ See Figure 4(b). (a) Express the area A of the opening as a function of .θ (b) Use a graphing utility to graph A A . θ( ) = Find the angle θ that makes A largest. (This bend will allow the most water to flow through the gutter.) EXAMPLE 2 Solution (a) Look again at Figure 4(b). The area A of the opening is the sum of the areas of two congruent right triangles and one rectangle. Look at Figure 4(c), which shows the triangle on the right in Figure 4(b) redrawn. Note that θ θ θ θ = = = = a a b b cos 4 , so 4 cos sin 4 , so 4 sin The area of the triangle is θ θ θ θ = ⋅ ⋅ = = ⋅ ⋅ = ab area 1 2 base height 1 2 1 2 4 cos 4 sin 8 sin cos So the area of the two congruent triangles together is θ θ θ θ ⋅ = 2 8sin cos 16sin cos . The rectangle has length 4 and height b, so its area is b area of rectangle 4 4 4sin 16sin θ θ = = ⋅ = (continued)
RkJQdWJsaXNoZXIy NjM5ODQ=