Chapter 5

Find the exact value. (a) \(\csc (3 \pi / 4)\) (b) \(\csc (-2 \pi / 3)\)

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\cos \theta=\frac{8}{17}$$

Exer. \(17-20\) : Express \(\theta\) in terms of degrees, minutes, and seconds, to the nearest second. $$\theta=1.5$$

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$\beta, b ; a$$

Use a formula for negatives to find the exact value. $$\text { (a) } \cot \left(-\frac{3 \pi}{4}\right) \quad \text { (b) } \sec \left(-180^{\circ}\right) \quad \text { (c) } \csc \left(-\frac{3 \pi}{2}\right)$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\sin \left(\frac{1}{2} x-\frac{\pi}{3}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\cot \left(x-\frac{\pi}{2}\right)$$

Approximate to three decimal places. (a) \(\sin 73^{\circ} 20^{\prime}\) (b) \(\cos 0.68\)

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\tan \theta=\frac{5}{12}$$

Exer. \(17-20\) : Express \(\theta\) in terms of degrees, minutes, and seconds, to the nearest second. $$\theta=5$$

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$\alpha, b ; a$$

Use a formula for negatives to find the exact value. $$\text { (a) } \cot \left(-225^{\circ}\right) \quad \text { (b) } \sec \left(-\frac{\pi}{4}\right) \quad \text { (c) } \csc \left(-45^{\circ}\right)$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\sin \left(\frac{1}{2} x+\frac{\pi}{4}\right)\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\cot \left(x+\frac{\pi}{4}\right)$$

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\cot \theta=\frac{7}{24}$$

Verify the identity by transforming the left hand side into the right-hand side. $$\sin (-x) \sec (-x)=-\tan x$$

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$\alpha, a ; \quad c$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=6 \sin \pi x\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\cot 2 x$$

Approximate to three decimal places. (a) \(\tan 21^{\circ} 10^{\prime} \) b) \(\cot 1.13\)

Exer. \(21-24:\) Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$37^{\circ} 41^{\prime}$$

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\sec \theta=\frac{6}{5}$$

Verify the identity by transforming the left hand side into the right-hand side. $$\csc (-x) \cos (-x)=-\cot x$$

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$\boldsymbol{\beta}, \boldsymbol{a} ; \boldsymbol{c}$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=3 \cos \frac{\pi}{2} x\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\cot \frac{1}{2} x$$

Approximate to three decimal places. (a) cot \(9^{\circ} 10^{\prime}\) b) \(\tan 0.75\)

Exer. \(21-24\) : Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$83^{\circ} 17^{\prime}$$

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\csc \theta=4$$

Verify the identity by transforming the left hand side into the right-hand side. $$\frac{\cot (-x)}{\csc (-x)}=\cos x$$

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$a, c ; \quad b$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=2 \cos \frac{\pi}{2} x\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\cot \frac{1}{3} x$$

Exer. \(21-24\) : Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$115^{\circ} 26^{\prime} 27^{\prime \prime}$$

A forester, 200 feet from the base of a redwood tree, observes that the angle between the ground and the top of the tree is \(60^{\circ} .\) Estimate the height of the tree.

Verify the identity by transforming the left hand side into the right-hand side. $$\frac{\sec (-x)}{\tan (-x)}=-\csc x$$

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$a, b ; \quad c$$

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=4 \sin 3 \pi x\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\cot 3 x$$

Approximate to three decimal places. (a) csc \(43^{\circ} 40^{\prime}\) (b) sec 0.26

The peak of Mt. Fuji in Japan is approximately \(12,400\) feet high. A trigonometry student, several miles away, notes that the angle between level ground and the peak is \(30^{\circ} .\) Estimate the distance from the student to the point on level ground directly beneath the peak.

Verify the identity by transforming the left hand side into the right-hand side. $$\frac{1}{\cos (-x)}-\tan (-x) \sin (-x)=\cos x$$

A person flying a kite holds the string 4 feet above ground level. The string of the kite is taut and makes an angle of \(60^{\circ}\) with the horizontal (see the figure). Approximate the height of the kite above level ground if 500 feet of string is payed out. (IMAGE CAN NOT COPY)

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\frac{1}{2} \sin 2 \pi x\)

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=2 \cot \left(2 x+\frac{\pi}{2}\right)$$

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\cos \theta=0.8620$$

Stonehenge in Salisbury Plains, England, was constructed using solid stone blocks weighing over \(99,000\) pounds each. Lifting a single stone required 550 people, who pulled the stone up a ramp inclined at an angle of \(9^{\circ} .\) Approximate the distance that a stone was moved in order to raise it to a height of 30 feet.

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$63.169^{\circ}$$

Verify the identity by transforming the left hand side into the right-hand side. $$\cot (-x) \cos (-x)+\sin (-x)=-\csc x$$

From a point 15 meters above level ground, a surveyor measures the angle of depression of an object on the ground at \(68^{\circ}\). Approximate the distance from the object to the point on the ground directly beneath the surveyor.