SECTION 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves 583 62. Show that the area K of triangle PQR is = K rs, where ( ) = + + s a b c 1 2 . Then show that ( )( )( ) = − − − r s a s b s c s 60. Use the result of Problem 59 and the results of Problems 62 and 63 in Section 8.3 to show that = − C s c r cot 2 where ( ) = + + s a b c 1 2 . 61. Show that + + = A B C s r cot 2 cot 2 cot 2 ‘Are You Prepared?’ Answers 1. = K bh 1 2 2. False Explaining Concepts 63. What do you do first if you are asked to find the area of a triangle and are given two sides and the included angle? 64. What do you do first if you are asked to find the area of a triangle and are given three sides? 65. State the formula for finding the area of an SAS triangle in words. Retain Your Knowledge Problems 66–75 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. 66. Without graphing, determine whether the quadratic function ( ) = − + + f x x x 3 12 5 2 has a maximum value or a minimum value, and then find the value. 67. Solve the inequality: + − ≤ x x 1 9 0 2 68. = − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ P 7 3 , 2 3 is the point on the unit circle that corresponds to a real number t. Find the exact values of the six trigonometric functions of t. 69. Establish the identity: θ θ θ θ − = csc sin cos cot 70. Find the domain of ( ) ( ) = − + f x x ln 25 3. 2 71. A rectangle has a diagonal of length 12. Express the perimeter P as a function of its width, w. 72. List all potential rational zeros of ( ) = − + + P x x x x 2 5 13 6. 3 2 73. Solve: ( ) − − ≤ x5 7 5 0.05 74. Solve: ( ) − = x x 7 18 75. The slope m of the tangent line to the graph of ( ) = − + f x x x 3 7 2 4 2 at any number x is given by ( ) = ′ = − m f x x x 12 14 . 3 Find an equation of the tangent line at = x 1. 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves Now Work the ‘Are You Prepared?’ problems on page 589. • Sinusoidal Graphs (Section 6.4, pp. 427–436) • Angular Speed (Section 6.1, pp. 390–391) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Build a Model for an Object in Simple Harmonic Motion (p. 583) 2 Analyze Simple Harmonic Motion (p. 585) 3 Analyze an Object in Damped Motion (p. 586) 4 Graph the Sum of Two Functions (p. 588) 1 Build a Model for an Object in Simple Harmonic Motion Many physical phenomena can be described as simple harmonic motion. Radio and television waves, light waves, sound waves, and water waves exhibit motion that is simple harmonic.
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