SECTION 6.2 Trigonometric Functions: Unit Circle Approach 411 135. Projectile Motion An object is propelled upward at an angle θ, θ ° < < ° 45 90 , to the horizontal with an initial velocity of v0 feet per second from the base of an inclined plane that makes an angle of 45° with the horizontal. See the figure. R 458 u If air resistance is ignored, the distance R that it travels up the inclined plane as a function of θ is given by θ θ θ ( ) ( ) ( ) [ ] = − − R v 2 32 sin 2 cos 2 1 0 2 (a) Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and 60 . θ = ° (b) Graph R R θ( ) = if the initial velocity is 32 feet per second. (c) What value of θ makes R largest? 136. If , 0 , θ θ π < < is the angle between the positive x-axis and a nonhorizontal, nonvertical line L, show that the slope m of L equals tan .θ The angle θ is called the inclination of L. y x M L O 21 1 (cos u, sin u) 21 1 u u [Hint: See the figure, where we have drawn the line M parallel to L and passing through the origin. Use the fact that M intersects the unit circle at the point cos , sin . θ θ ( ) ] In Problems 137 and 138, use the figure to approximate the value of the six trigonometric functions at t to the nearest tenth. Then use a calculator to approximate each of the six trigonometric functions at t. 0.5 20.5 20.5 0.5 2 1 6 5 4 3 Unit Circle y x 137. (a) t 1 = (b) t 5.1 = 138. (a) t 2 = (b) t 4 = 139. Challenge Problem Let θ be the measure of an angle, in radians, in standard position with 3 2 . π θ π < < Find the exact y-coordinate of the intersection of the terminal side of θ with the unit circle, given cos sin 41 49 . 2 θ θ + = State the answer as a single fraction, completely simplified, with rationalized denominator. a cone of height h and radius r. See the figure. Many cones can be used to enclose the sphere, each having a different slant angle θ. The volume V of the cone can be expressed as a function of the slant angle θ of the cone as θ π θ θ θ ( ) ( ) ( ) = + ° < < ° V R 1 3 1 sec tan 0 90 3 3 2 What volume V is required to enclose a sphere of radius 2 centimeters in a cone whose slant angle θ is 30°? 45°? 60°? O r R h u Use the following to answer Problems 129–132.The viewing angle, ,θ of an object is the angle the object forms at the lens of the viewer’s eye.This is also known as the perceived or angular size of the object. The viewing angle is related to the object’s height, H, and distance from the viewer, D, through the formula H D tan 2 2 . θ = 129. Tailgating While driving, Arletha observes the car in front of her with a viewing angle of 22 .° If the car is 6 feet wide, how close is Arletha to the car in front of her? Round your answer to one decimal place. 130. Viewing Distance The Washington Monument in Washington, D.C. is 555 feet tall. If a tourist sees the monument with a viewing angle of 8 ,° how far away, to the nearest foot, is she from the monument? 131. Tree Height A forest ranger views a tree that is 200 feet away with a viewing angle of 20 .° How tall is the tree to the nearest foot? 132. Radius of the Moon An astronomer observes the moon with a viewing angle of 0.52 .° If the moon’s average distance from Earth is 384,400 km, what is its radius to the nearest kilometer? 133. Projectile Distance An object is fired at an angle θ to the horizontal with an initial speed of v0 feet per second. Ignoring air resistance, the length of the projectile’s path is given by L v 32 sin cos ln tan 2 4 0 2 2 θ θ θ π θ ( ) ( ) ( ) ( ) = − ⋅ ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ where 0 2 . θ π < < (a) Find the length of the object’s path for angles 6 , 4 , θ π π = and 3 π if the initial velocity is 128 feet per second. (b) Using a graphing utility, determine the angle required for the object to have a path length of 550 feet if the initial velocity is 128 feet per second. (c) What angle will result in the longest path? How does this angle compare to the angle that results in the longest range? (See Problems 121–124.) 134. Photography The length L of the chord joining the endpoints of an arc on a circle of radius r subtended by a central angle θ, θ π < ≤ 0 , is given by L r 2 2cos .θ = − Use this fact to approximate the field width (the width of scenery the lens can image) of a 450 mm camera lens at a distance of 920 feet if the viewing angle of the lens is 30 . π
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