546 CHAPTER 8 Applications of Trigonometric Functions Brian May (1947-present) Credit: Pictorial Press Ltd/Alamy Stock Photo Brian May is the lead guitarist of the rock band Queen. Brian May’s strength in mathematics had him on a path to be an astrophysicist before he ended up joining the band and becoming a full-time musician. Years later, May returned to complete and publish his doctoral thesis. Recently, he helped advise NASA on the New Horizons mission studying Pluto. 8.1 Right Triangle Trigonometry; Applications • Pythagorean Theorem (Section A.2, pp. A14–A15) • Trigonometric Equations (Section 7.3, pp. 493–498) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles (p. 546) 2 Use the Complementary Angle Theorem (p. 548) 3 Solve Right Triangles (p. 548) 4 Solve Applied Problems (p. 549) Now Work the ‘Are You Prepared?’ problems on page 553. 1 Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles A triangle in which one angle is a right angle 90 ( )° is called a right triangle . Recall that the side opposite the right angle is called the hypotenuse , and the remaining two sides are called the legs of the triangle. In Figure 1(a), the hypotenuse is labeled as c to indicate that its length is c units, and, in a like manner, the legs are labeled as a and b. Because the triangle is a right triangle, the Pythagorean Theorem tells us that a b c 2 2 2 + = Figure 1(a) also shows the angle .θ The angle θ is an acute angle : that is, 0 90 θ ° < < ° for θ measured in degrees and 0 2 θ π < < for θ measured in radians. Place θ in standard position, as shown in Figure 1(b). Then the coordinates of the point P are a b , . ( ) Also, P is a point on the terminal side of θ that is on the circle x y c . 2 2 2 + = (Do you see why?) Figure 1 Right triangle with acute angle θ a b c Hypotenuse (a) u a b c y x P 5 (a, b) O u (b) x2 1 y2 5 c2 Now apply the theorem on page 407 for evaluating trigonometric functions using a circle of radius c x y c , . 2 2 2 + = By referring to the lengths of the sides of the triangle by the names hypotenuse c , ( ) opposite b , ( ) and adjacent a , ( ) as indicated in Figure 2, the trigonometric functions of θ can be expressed as ratios of the sides of a right triangle. Figure 2 Right triangle c b a Hypotenuse Opposite u Adjacent to u u Notice that each trigonometric function of the acute angle θ is positive. b c c b a c c a b a a b •sin Opposite Hypotenuse •csc Hypotenuse Opposite •cos Adjacent Hypotenuse •sec Hypotenuse Adjacent • tan Opposite Adjacent •cot Adjacent Opposite θ θ θ θ θ θ = = = = = = = = = = = = (1)
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