Chapter 5

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. I; on a line having slope \(\frac{4}{3}\)

Graph \(f\) on the Interval \([-2 \pi, 2 \pi],\) and estimate the coordinates of the high and low points. $$f(x)=\sin ^{2} x \cos x$$

Use a graph to estimate the largest interval \([a, b],\) with \(a<0\) and \(b>0,\) on which \(f\) is one-to-one. $$f(x)=1.5 \cos \left(\frac{1}{2} x-0.3\right)+\sin (1.5 x+0.5)$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. III; bisects the quadrant

Use a graph to solve the inequality on the interval \([-\pi, \pi]\) $$\cos (2 x-1)+\sin 3 x \geq \sin \frac{1}{3} x+\cos x$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. III; parallel to the line \(2 y-7 x+2=0\)

As \(x \rightarrow 0^{+}, f(x) \rightarrow L\) for some real number \(L\) Use a graph to predict \(L\) $$f(x)=\frac{6 x-6 \sin x}{x^{3}}$$

Use a graph to solve the inequality on the interval \([-\pi, \pi]\) $$\frac{1}{2} \cos 2 x+2 \cos (x-2)<\cos (1.5 x+1)+\sin (x-1)$$

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. II; parallel to the line through \(A(1,4)\) and \(B(3,-2)\)

Radio signal intensity Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength \(\lambda\) and the dis. tance between the two radio towers is equal to \(\frac{1}{2} \lambda\), then the intensity \(I\) of the signal in the direction \(\theta\) is given by $$I=\frac{1}{2} I_{0}[1+\cos (\pi \sin \theta)]$$ where \(I_{0}\) is the maximum intensity. Approximate \(I\) in terms of \(I_{0}\) for each \(\theta .\) (a) \(\theta=0\) (b) \(\quad \theta=\pi / 3\) (c) \(\theta=\pi / 7\)

Find the exact values of the six trigonometric functions of each angle, whenever possible. (a) \(90^{\circ}\) (6) \(0^{\circ}\) (c) \(7 \pi / 2\) (d) \(3 \pi\)

As \(x \rightarrow 0^{+}, f(x) \rightarrow L\) for some real number \(L\) Use a graph to predict \(L\) $$f(x)=\frac{x+\tan x}{\sin x}$$

Find the exact values of the six trigonometric functions of each angle, whenever possible. (a) \(180^{\circ}\) (b) \(-90^{\circ}\) (c) \(2 \pi\) (d) \(5 \pi / 2\)

Find the quadrant containing \(\theta\) if the given conditions are true. (a) \(\cos \theta>0\) and \(\sin \theta<0\) (b) \(\sin \theta<0\) and \(\cot \theta>0\) (c) \(\csc \theta>0\) and \(\sec \theta<0\) (d) \(\sec \theta<0\) and \(\tan \theta>0\)

Find the quadrant containing \(\theta\) if the given conditions are true. (a) \(\tan \theta<0\) and \(\cos \theta>0\) (b) sec \(\theta>0\) and \(\tan \theta<0\) (c) \(\csc \theta>0\) and \(\cot \theta<0\) (d) \(\cos \theta<0\) and \(\csc \theta<0\)

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\tan \theta=-\frac{3}{4} \text { and } \sin \theta>0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\cot \theta=\frac{3}{4} \text { and } \cos \theta<0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\sin \theta=-\frac{5}{13} \text { and } \sec \theta>0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\cos \theta=\frac{1}{2} \text { and } \sin \theta<0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\cos \theta=-\frac{1}{3} \text { and } \sin \theta<0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\csc \theta=5 \text { and } \cot \theta<0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\sec \theta=-4 \text { and } \csc \theta>0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\sin \theta=\frac{2}{5} \text { and } \cos \theta<0$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{\sec ^{2} \theta-1} ; \quad \pi / 2<\theta<\pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{1+\cot ^{2} \theta}, \quad 0<\theta<\pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{1+\tan ^{2} \theta} ; \quad 3 \pi / 2<\theta<2 \pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{\csc ^{2} \theta-1} ; \quad 3 \pi / 2<\theta<2 \pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{\sin ^{2}(\theta / 2)} ; \quad 2 \pi<\theta<4 \pi$$

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{\cos ^{2}(\theta / 2)} ; \quad 0<\theta<\pi$$