426 CHAPTER 6 Trigonometric Functions 128. Show that the period of θ θ ( ) = f csc is π2 . 129. Show that the period of θ θ ( ) = f tan is π. 130. Show that the period of θ θ ( ) = f cot is π. 131. Prove the reciprocal identities given in identity (2). 132. Prove the quotient identities given in identity (3). 133. Establish the identity: θ φ θ φ θ ( ) ( ) + + = sin cos sin sin cos 1 2 2 2 134. Challenge Problem If θ θ θ θ + = + 2sin 3cos 3sin cos 1 2 2 with θ in quadrant I, find the possible values for θ cot . 135. Challenge Problem If θ θ = − tan 3 sec with θ in quadrant I, what is θ θ + sin cos ? 136. Challenge Problem If θ θ ( ) ( ) = sin 4 cos2 and θ π < < 0 4 2 , find the exact value of θ θ ( ) ( ) + − sin 8 cot 4 2. 137. Challenge Problem Find the exact value of θ θ − sin cos if θ θ − = cos 8sin 7 and θ ° < < ° 180 270 . 122. Lung Volume Normal resting lung volume V, in mL, for adult men varies over the breathing cycle and can be approximated by the model π ( ) ( ) = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ + V t t 250 sin 2 1.25 5 2650 where t is the number of seconds after breathing begins. Use the model to estimate the volume of air in a man’s lungs after 2.5 seconds, 10 seconds, and 17 seconds. 123. Show that the range of the tangent function is the set of all real numbers. 124. Show that the range of the cotangent function is the set of all real numbers. 125. Show that the period of θ θ ( ) = f sin is π2 . [Hint: Assume that π < < p 0 2 exists so that θ θ ( ) + = p sin sin for all θ. Let θ = 0 to find p. Then let θ π = 2 to obtain a contradiction.] 126. Show that the period of θ θ ( ) = f cos is π2 . 127. Show that the period of θ θ ( ) = f sec is π2 . Explaining Concepts 138. Write down five properties of the tangent function. Explain the meaning of each. 139. Describe your understanding of the meaning of a periodic function. 140. Explain how to find the value of sin ° 390 using periodic properties. 141. Explain how to find the value of ( ) − ° cos 45 using even-odd properties. 142. Explain how to find the value of ° sin390 and ( ) − ° cos 45 using the unit circle. ‘Are You Prepared?’ Answers 1. { } ≠ − x x 1 2 2. even 3. False 4. True Problems 143–152 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. Retain Your Knowledge 143. Given: ( ) = − f x x 3 2 and ( ) = − g x x 7, find ( )( ) f g x . 144. Graph ( ) = − + − f x x x 2 12 13 2 using transformations. Find the vertex and the axis of symmetry. 145. Solve exactly: = − e 6 x 4 146. Find the real zeros of ( ) = − + − f x x x x 9 3 27. 3 2 147. Solve: + − − = x x 2 5 2 148. If the real zeros of ( ) g x are −2 and 3, what are the real zeros of ( ) + g x 6 ? 149. Solve: ( ) − = x log 5 2 4 150. Find c so that ( ) = − + f x x x c 6 28 2 has a minimum value of 7 3 . 151. Find the intercepts of the graph of − + = x y 3 5 15. 152. Find the difference quotient for ( ) = − + f x x x 3 2 5 1 2 .
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