SECTION 6.3 Properties of the Trigonometric Functions 425 109. Is the tangent function even, odd, or neither? Is its graph symmetric? With respect to what? 110. Is the cotangent function even, odd, or neither? Is its graph symmetric? With respect to what? 111. Is the secant function even, odd, or neither? Is its graph symmetric? With respect to what? 112. Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what? 104. What is the range of the cotangent function? 105. What is the range of the secant function? 106. What is the range of the cosecant function? 107. Is the sine function even, odd, or neither? Is its graph symmetric? With respect to what? 108. Is the cosine function even, odd, or neither? Is its graph symmetric? With respect to what? Applications and Extensions In Problems 113–118, use the periodic and even-odd properties. 113. If θ θ ( ) = f sin and ( ) = f a 1 3 , find the exact value of: (a) ( ) − f a (b) π π ( ) ( ) ( ) + + + + f a f a f a 2 4 114. If θ θ ( ) = f cos and ( ) = f a 1 4 , find the exact value of: (a) ( ) − f a (b) π π ( ) ( ) ( ) + + + − f a f a f a 2 2 115. If θ θ ( ) = f tan and ( ) = f a 2, find the exact value of: (a) ( ) − f a (b) π π ( ) ( ) ( ) + + + + f a f a f a 2 116. If θ θ ( ) = f cot and ( ) = − f a 3, find the exact value of: (a) ( ) − f a (b) π π ( ) ( ) ( ) + + + + f a f a f a 4 117. If θ θ ( ) = f sec and ( ) = − f a 4, find the exact value of: (a) ( ) − f a (b) π π ( ) ( ) ( ) + + + + f a f a f a 2 4 118. If θ θ ( ) = f csc and ( ) = f a 2, find the exact value of: (a) ( ) − f a (b) π π ( ) ( ) ( ) + + + + f a f a f a 2 4 119. Calculating the Time of a Trip From a parking lot, you want to walk to a house on the beach. The house is located 1500 feet down a paved path that parallels the ocean, which is 500 feet away. See the figure. Along the path you can walk 300 feet per minute, but in the sand on the beach you can only walk 100 feet per minute. The time T to get from the parking lot to the beach house expressed as a function of the angle θ shown in the figure is θ θ θ θ π ( ) = − + < < T 5 5 3tan 5 sin , 0 2 Calculate the time T if you walk directly from the parking lot to the house. [Hint: θ = tan 500 1500 .] Beach Ocean Paved path Parking lot u x 500 ft 1500 ft Forest 120. Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. Sally can jog 8 miles per hour on the paved path, but only 3 miles per hour in the sand on the beach. Because a river flows directly between the two houses, it is necessary to jog in the sand to the road, continue on the path, and then jog directly back in the sand to get from one house to the other. See the figure.The time T to get from one house to the other as a function of the angle θ shown in the figure is θ θ θ θ π ( ) = + − < < T 1 2 3sin 1 4 tan , 0 2 (a) Calculate the time T for θ = tan 1 4 . (b) Describe the path taken. (c) Explain why θ must be larger than ° 14 . 121. Predator Population In predator–prey relationships, the populations of the predator and prey are often cyclical. In a conservation area, rangers monitor the red fox population and have determined that the population can be modeled by the function π( ) ( ) = + P t t 40cos 6 110 where t is the number of months from the time monitoring began. Use the model to estimate the population of red foxes in the conservation area after 10 months, 20 months, and 30 months. Credit: Eric Isselee/Shutterstock Ocean Beach Paved path River 1 mi x 4 mi 4 mi u u
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