Chapter 6

Exer. 5-40: 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. 17-24: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), express the third part in terms of the first two. $$ \alpha, b ; \quad a $$

Exer. 5-40: 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. $$ y=\cot \left(x+\frac{\pi}{4}\right) $$

Approximate to three decimal places. (a) \(\cos 38^{\circ} 30^{\prime}\) (b) \(\sin 1.48\)

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

Express \(\theta\) in terms of degrees, minutes, and seconds, to the nearest second. $$\theta=4$$

Exer. 17-24: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), express the third part in terms of the first two. $$ \alpha, a ; \quad c $$

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \sin (-x) \sec (-x)=-\tan x $$

Exer. 5-40: 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) \(\tan 21^{\circ} 10^{r}\) (b) \(\cot 1.13\)

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

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

Exer. 17-24: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), express the third part in terms of the first two. $$ \beta, a ; \quad c $$

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \csc (-x) \cos (-x)=-\cot x $$

Exer. 5-40: 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. $$ y=\cot \frac{1}{2} x $$

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

Exer. 17-24: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), express the third part in terms of the first two. $$ a, c ; \quad b $$

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \frac{\cot (-x)}{\csc (-x)}=\cos x $$

Exer. 5-40: 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. $$ y=\cot \frac{1}{3} x $$

Approximate to three decimal places. (a) \(\sec 67^{\circ} 50^{\prime}\) (b) \(\csc 0.32\)

Height of a tree 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.

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

Exer. 17-24: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), express the third part in terms of the first two. $$ a, b ; \quad c $$

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \frac{\sec (-x)}{\tan (-x)}=-\csc x $$

Exer. 5-40: 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) \(\sec 0.26\)

Distance to Mt. Fuji 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.

Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$258^{\circ} 39^{\prime} 52^{\prime \prime}$$

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \frac{1}{\cos (-x)}-\tan (-x) \sin (-x)=\cos x $$

Exer. 5-40: 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 blocks 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.

Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$63.169^{\circ}$$

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.

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \cot (-x) \cos (-x)+\sin (-x)=-\csc x $$

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\frac{1}{2} \cos \frac{\pi}{2} x $$

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

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\). $$ \sin \theta=0.6612 $$

Advertising sign height Added in 1990 and removed in 1997 , the highest advertising sign in the world was a large letter I situated at the top of the 73-story First Interstate World Center building in Los Angeles. At a distance of 200 feet from a point directly below the sign, the angle between the ground and the top of the sign was \(78.87^{\circ}\). Approximate the height of the top of the sign.

Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$12.864^{\circ}$$

A pilot, flying at an altitude of 5000 feet, wishes to approach the numbers on a runway at an angle of \(10^{\circ}\). Approximate, to the nearest 100 feet, the distance from the airplane to the numbers at the beginning of the descent.