Chapter 4

In Problems \(1-12,\) use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ \cos 4 \theta \cos 3 \theta $$

Complete the given table. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & \frac{2 \pi}{3} & \frac{3 \pi}{4} & \frac{5 \pi}{6} & \pi & \frac{7 \pi}{6} & \frac{5 \pi}{4} & \frac{4 \pi}{3} & \frac{3 \pi}{2} & \frac{5 \pi}{3} & \frac{7 \pi}{4} & \frac{11 \pi}{6} & 2 \pi \\ \hline \tan x & & & & & & & & & & & & \\ \hline \cot x & & & & & & & & & & & & \\ \hline \end{array} $$

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=\frac{1}{2}+\cos x $$

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \sec t \cos t $$

In Problems \(1-6\), find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \sin x=\sqrt{3} / 2 $$

In Problems \(1-6,\) proceed as in Example 2 and reduce the given trigonometric expression to the form \(y=A\) \(\sin (B x+\phi)\). Sketch the graph and give the amplitude, the period, and the phase shift. \(y=\cos \pi x-\sin \pi x\)

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ 60^{\circ} $$

Given that \(\cos t=-\frac{2}{5}\) and that \(P(t)\) is a point in the second quadrant, find sin \(t\).

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \cos \frac{\pi}{12} $$

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ \sin \frac{3 t}{2} \cos \frac{t}{2} $$

Complete the given table. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline x & \frac{2 \pi}{3} & \frac{3 \pi}{4} & \frac{5 \pi}{6} & \pi & \frac{7 \pi}{6} & \frac{5 \pi}{4} & \frac{4 \pi}{3} & \frac{3 \pi}{2} & \frac{5 \pi}{3} & \frac{7 \pi}{4} & \frac{11 \pi}{6} & 2 \pi \\ \hline \sec x & & & & & & & & & & & & \\ \hline \csc x & & & & & & & & & & & & \\ \hline \end{array} $$

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=-1+\cos x $$

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \tan \alpha \cos \alpha $$

Find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \cos x=-\sqrt{2} / 2 $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \tan ^{-1} \sqrt{3} $$

Proceed as in Example 2 and reduce the given trigonometric expression to the form \(y=A\) \(\sin (B x+\phi)\). Sketch the graph and give the amplitude, the period, and the phase shift. \(y=\sin \frac{\pi}{2} x-\sqrt{3} \cos \frac{\pi}{2} x\)

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ -120^{\circ} $$

Given that \(\sin t=\frac{1}{4}\) and that \(P(t)\) is a point in the second quadrant, find \(\cos t\)

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin \frac{\pi}{12} $$

Find the indicated value without the use of a calculator. $$ \cot \frac{13 \pi}{6} $$

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=2-\sin x $$

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \frac{\sin \theta}{\csc \theta}+\frac{\cos \theta}{\sec \theta} $$

Find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \sec x=\sqrt{2} $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arccos (-1) $$

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ 135^{\circ} $$

Given that \(\sin t=-\frac{2}{3}\) and that \(P(t)\) is a point in the third quadrant, find \(\cos t\).

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin 75^{\circ} $$

Find the indicated value without the use of a calculator. $$ \csc \left(-\frac{3 \pi}{2}\right) $$

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=3+3 \sin x $$

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \frac{\csc ^{2} x-1}{\cot x} $$

Find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \tan x=-1 $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arcsin \frac{\sqrt{3}}{2} $$

Proceed as in Example 2 and reduce the given trigonometric expression to the form \(y=A\) \(\sin (B x+\phi)\). Sketch the graph and give the amplitude, the period, and the phase shift. \(y=\sqrt{3} \cos 4 x-\sin 4 x\)

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ 150^{\circ} $$

Given that \(\cos t=\frac{3}{4}\) and that \(P(t)\) is a point in the fourth quadrant, find \(\sin t\)

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \cos 75^{\circ} $$

Find the indicated value without the use of a calculator. $$ \tan \frac{9 \pi}{2} $$

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=-2+4 \cos x $$

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \tan ^{2} t-\sec ^{2} t $$

Find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \cot x=-\sqrt{3} $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arccos \frac{1}{2} $$

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ 1140^{\circ} $$

If \(\sin t=-\frac{2}{7}\), find all possible values of \(\cos t\).

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin \frac{17 \pi}{12} $$

Find the indicated value without the use of a calculator. $$ \sec 7 \pi $$

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=1-2 \sin x $$

Use the fundamental identities and the even-odd identities to simplify each expression. $$ 1+\tan ^{2}(-\theta) $$

Find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \csc x=2 $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arctan (-\sqrt{3}) $$

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ -315^{\circ} $$