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Section 1.4: Trig Functions and Their Inverses Homework Questions π : 6 π 1 Solution: sin = 6 2 1. Evaluate sin 5π 2. Evaluate: tan − 6 Solution: 5π 5π sin sin − 5π 6 6 =− tan − = 5π 5π 6 cos − cos 6 6 1/2 =− √ − 3/2 √ 3 = 3 3. Establish the identity (1 + cot2 θ) sin2 θ = 1 Solution: (1 + cot2 θ) sin2 θ = sin2 θ cot2 θ sin2 θ cos2 θ sin2 θ sin2 θ = sin2 θ + cos2 θ = 1 = sin2 θ + 4. Find the exact value of sin−1 (1) √ ! 3 2 Solution: The question is asking at what angle θ is sin θ = 1 √ 3/2?. We want to solve −1 sin √ ! 3 = θ for θ. This gives 2 √ !! 3 = sin θ 2 −1 sin sin or √ sin θ = π 3 so θ = 2 3 5. Find the exact value of sec−1 (−1). Solution: We know that sec θ = sec −1 1 , so cos θ −1 (−1) = cos 1 −1 =π 6. Use a right triangle to write cos(sin−1 (5x)) as an algebraic expression. Assume x is positive and that the given inverse trigonometric is defined for x. Solution: We know that sin θ = 5x. The corresponding right triangle has opposite side length 5x and hypotenuse length 1. By the Pythagorean Theorem, √ the adjacent side has length 1 − 25x2 . We know that cos θ is Adjacent over Hypotenuse, giving √ 1 − 25x2 −1 cos(sin (5x)) = 1 7. Find the exact value for the remaining trigonometric functions of θ given cos θ = − 24 , θ in quadrant III 25 Solution θ corresponds to the following right triangle: 2 but is in the third quadrant. So, sin θ = − 7 25 7 24 24 cot θ = 7 25 sec θ = − 24 25 csc θ = − 7 tan θ = 8. Determine the amplitude and period of y = 5.4 cos πx 9 2π = 18 Solution: Amplitude = 5.4. Period = πx/9 √ 9. Solve the equation 2 cos x − 2 = 0 for all solutions of x. √ √ 2 Solution: 2 cos x − 2 = 0 gives cos x = or 2 π x = + 2πk, 4 x= 7π + 2πk 4 for all k = 0, 1, 2, . . . 10. Two ladders of length a lean against opposite walls of an alley with their feet touching (see figure). One ladder extends h feet up the wall and makes a 75◦ angle with the ground. The other ladder extends k feet up the opposite wall and makes a 45◦ angle with the ground. Find the width of the alley. 3 Solution: Width = a cos(75◦ ) + a cos(45◦ ) 4