462 CHAPTER 6 Trigonometric Functions Retain Your Knowledge Problems 42–51 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 42. Given ( ) ( ) = + − f x x f x 4 9 2 , find . 1 43. Solve: ( ) + = − x x 0.25 0.4 0.8 3.7 1.4 44. Multiply: ( ) +x y 8 15 2 45. Find the exact distance between the points ( ) ( ) − 4, 1 and 10,3 . 46. Solve: + = − x x 3 4 5 7 47. Given = + y x x 4, let = + u x 4 and express y in terms of u. 48. If = > x a t a sin , 0, and π π − ≤ ≤ t 2 2 , find cos t. 49. A rectangular garden is enclosed by 54 feet of fencing. If the length of the garden is 3 feet more than twice the width, what are the dimensions of the garden? 50. Find the vertical asymptotes, if any, of the graph of ( ) = − − − R x x x x 25 2 15 . 2 2 51. Write ( ) x y log 8 2 2 5 as a sum of logarithms. Express powers as factors. Chapter Review Things to Know Definitions Angle in standard position (p. 383) Vertex is at the origin; initial side is along the positive x-axis. 1 Degree (1º) (p. 384) 1 1 360 ° = revolution 1 Radian (p. 386) The measure of a central angle of a circle whose rays subtend an arc whose length is equal to the radius of the circle. Trigonometric functions (p. 398) P x y , ( ) = is the point on the unit circle corresponding to t θ = radians. θ θ θ = = = = = = ≠ t y t x t y x x sin sin cos cos tan tan , 0 θ θ θ = = ≠ = = ≠ = = ≠ t y y t x x t x y y csc csc 1 , 0 sec sec 1 , 0 cot cot , 0 Trigonometric functions using a circle of radius r (pp. 406–407) For an angle θ in standard position, ( ) = P x y , is the point on the terminal side of θ that is also on the circle + = x y r . 2 2 2 θ θ θ θ θ θ = = = ≠ = ≠ = ≠ = ≠ y r x r y x x r y y r x x x y y sin cos tan , 0 csc , 0 sec , 0 cot , 0 Periodic function (p. 415) A function f is periodic if for some number > p 0 for which θ + p is in the domain of f whenever θ is, then θ θ ( ) ( ) + = f p f . The smallest such p is the fundamental period. Formulas 1 counterclockwise π ° = 1 180 radian (p. 388); π = 1 radian 180 degrees (p. 388) = ° revolution 360 (p. 385) = π2 radians (p. 388) Arc length: θ = s r (p. 386) θ is measured in radians; s is the length of the arc subtended by the central angle θ of the circle of radius r. Area of a sector: θ = A r 1 2 2 (p. 390) A is the area of the sector of a circle of radius r formed by a central angle of θ radians.
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