SECTION 7.1 The Inverse Sine, Cosine, and Tangent Functions 485 (a) What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet? (b) If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row behind it, which row results in the largest viewing angle? (c) Using a graphing utility, graph θ ( ) ( ) ( ) = − − − x x x tan 34 tan 6 1 1 What value of x results in the largest viewing angle? 87. Area under a Curve The area under the graph of = + y x 1 1 2 and above the x-axis between = x a and = x b is given by − − − b a tan tan 1 1 See the figure. y 1.5 x 3 23 b a (a) Find the exact area under the graph of = + y x 1 1 2 and above the x-axis between = x 0 and = x 3. (b) Find the exact area under the graph of = + y x 1 1 2 and above the x-axis between = − x 3 3 and = x 1. 88. Area under a Curve The area under the graph of = − y x 1 1 2 and above the x-axis between = x a and = x b is given by − − − b a sin sin 1 1 See the figure. y 3 x x 5 21 x 5 1 b a 21 1 (a) Find the exact area under the graph of = − y x 1 1 2 and above the x-axis between = x 0 and = x 3 2 . (b) Find the exact area under the graph of = − y x 1 1 2 and above the x-axis between = − x 1 2 and = x 1 2 . 84. Approximate the number of hours of daylight for any location that is ° ′ 66 30 north latitude for the following dates: (a) Summer solstice ( ) = ° i 23.5 (b) Vernal equinox ( ) = ° i 0 (c) July 4 ( ) = ° ′ i 22 48 (d) Thanks to the symmetry of the orbital path of Earth around the Sun, the number of hours of daylight on the winter solstice may be found by computing the number of hours of daylight on the summer solstice and subtracting this result from 24 hours. Compute the number of hours of daylight for this location on the winter solstice. What do you conclude about daylight for a location at ° ′ 66 30 north latitude? 85. Being the First to See the Rising Sun Cadillac Mountain, elevation 1530 feet, is located in Acadia National Park, Maine, and is the highest peak on the east coast of the United States. It is said that a person standing on the summit will be the first person in the United States to see the rays of the rising Sun. How much sooner would a person atop Cadillac Mountain see the first rays than a person standing below, at sea level? 2710 miles Rotation of Earth P D Q s First rays Sun u [Hint: Consult the figure.When the person at D sees the first rays of the Sun, the person at P does not. The person at P sees the first rays of the Sun only after Earth has rotated so that P is at location Q. Compute the length of the arc subtended by the central angle θ. Then use the fact that at the latitude of Cadillac Mountain, in 24 hours a length of π ⋅ ≈ 2 2710 17027.4 miles is subtended.] 86. Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, θ is the viewing angle. Suppose that you sit x feet from the screen. The viewing angle θ is given by the function θ ( ) ( ) ( ) = − − − x x x tan 34 tan 6 1 1 6 feet 28 feet u
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