SECTION 7.2 The Inverse Trigonometric Functions (Continued) 487 7.2 The Inverse Trigonometric Functions (Continued) Now Work the ‘Are You Prepared?’ problems on page 490. • Finding Exact Values of the Trigonometric Functions, Given the Value of a Trigonometric Function and the Quadrant of the Angle (Section 6.3, pp. 419–422) • Graphs of the Secant, Cosecant, and Cotangent Functions (Section 6.5, pp. 445–448) • Domain and Range of the Secant, Cosecant, and Cotangent Functions (Section 6.3, pp. 413–414) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Define the Inverse Secant, Cosecant, and Cotangent Functions (p. 487) 2 Find the Value of Inverse Secant, Cosecant, and Cotangent Functions (p. 487) 3 Find the Exact Value of Composite Functions Involving the Inverse Trigonometric Functions (p. 489) 4 Write a Trigonometric Expression as an Algebraic Expression (p. 490) 1 Define the Inverse Secant, Cosecant, and Cotangent Functions The inverse secant, inverse cosecant, and inverse cotangent functions are defined as follows: DEFINITION Inverse Secant, Cosecant, and Cotangent Functions • y x x y sec if and only if sec 1 = = − x y y where 1 and 0 , 2 * π π ≥ ≤ ≤ ≠ (1) • y x x y csc if and only if csc 1 = = − π π ≥ − ≤ ≤ ≠ x y y where 1 and 2 2 , 0† (2) • y x x y cot if and only if cot 1 = = − π −∞< <∞ < < x y where and 0 (3) Take time to review the graphs of the cotangent, cosecant, and secant functions in Figures 69, 73, and 74 in Section 6.5 to see the basis for these definitions. Now Work PROBLEM 4 *Most texts use this definition. A few use the restriction y y 0 2 , 3 2 . π π π ≤ < ≤ < † Most texts use this definition. A few use the restriction y y 2 , 0 2 . π π π − < ≤− < ≤ Finding the Exact Value of Inverse Secant, Cosecant, and Cotangent Functions Find the exact value of each expression. (a) sec 11− (b) csc 21− (c) cot 3 1( ) − − EXAMPLE 1 Solution (a) Let sec 1. 1 θ = − Then θ is the angle, 0 , 2 , θ π θ π ≤ ≤ ≠ whose secant equals 1 (or, equivalently, whose cosine equals 1). sec 1 0 , 2 sec 1 0 , 2 1 θ θ π θ π θ θ π θ π = ≤ ≤ ≠ = ≤ ≤ ≠ − 2 Find the Value of Inverse Secant, Cosecant, and Cotangent Functions (continued)
RkJQdWJsaXNoZXIy NjM5ODQ=