Chapter 4

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

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. $$ -\frac{\pi}{6} $$

Find the exact value of (a) \(\sin t\) and (b) \(\cos t\) for the given value of \(t\). Do not use a calculator. $$ t=-3 \pi / 2 $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \cos 165^{\circ} $$

Find the indicated value without the use of a calculator. $$ \sec \frac{29 \pi}{4} $$

Reduce the given expression to a single trigonometric function. $$ \frac{\tan t+\cot t}{\csc t} $$

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

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arctan 0 $$

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. $$ -3 \pi $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin 165^{\circ} $$

Find the indicated value without the use of a calculator. $$ \sec \left(-120^{\circ}\right) $$

Reduce the given expression to a single trigonometric function. $$ \sin x+\cos x \cot x $$

Find all solutions of the given trigonometric equation if \(\theta\) represents an angle measured in degrees. $$ 1+\cot \theta=0 $$

The equation of simple harmonic motion of a spring/mass system is \(y(t)=\frac{5}{2} \sin (2 t-\pi / 3)\). Determine the initial displacement \(y_{0}\) and initial velocity \(v_{0}\) of the mass.

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \sin \left(\cos ^{-1} \frac{3}{5}\right) $$

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. $$ 3 $$

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=2 \pi / 3 $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \tan 165^{\circ} $$

Find the indicated value without the use of a calculator. $$ \tan 405^{\circ} $$

Reduce the given expression to a single trigonometric function. $$ \sin \theta \tan \theta \csc ^{2} \theta-\sin \theta \tan \theta $$

Find all solutions of the given trigonometric equation if \(\theta\) represents an angle measured in degrees. $$ \sqrt{3} \sin \theta=\cos \theta $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \cos \left(\sin ^{-1} \frac{1}{3}\right) $$

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. $$ 4 $$

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=4 \pi / 3 $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \cos 195^{\circ} $$

Find the indicated value without the use of a calculator. $$ \csc 495^{\circ} $$

Find the \(x\) -intercepts of the graph of the given function on the interval \([\mathrm{o}, 2 \pi]\). Then find all intercepts using periodicity. $$ y=-1+\sin x $$

Reduce the given expression to a single trigonometric function. $$ \frac{\sec ^{2} \alpha}{\cos \alpha+\cos \alpha \tan ^{2} \alpha} $$

Find all solutions of the given trigonometric equation if \(\theta\) represents an angle measured in degrees. $$ \sec \theta=-2 $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \tan \left(\arccos \left(-\frac{2}{3}\right)\right) $$

In Problems \(17-20,\) express the given angle in decimal notation. $$ 10^{\circ} 39^{\prime} 17^{\prime \prime} $$

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=5 \pi / 4 $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin 195^{\circ} $$

Find the indicated value without the use of a calculator. $$ \cot \left(-720^{\circ}\right) $$

Find the \(x\) -intercepts of the graph of the given function on the interval \([\mathrm{o}, 2 \pi]\). Then find all intercepts using periodicity. $$ y=1-2 \cos x $$

Reduce the given expression to a single trigonometric function. $$ \frac{\sin ^{2} \theta \cos \theta+\cos ^{3} \theta-\cos \theta+\sin \theta}{\cos \theta} $$

Find all solutions of the given trigonometric equation if \(\theta\) represents an angle measured in degrees. $$ 2 \cos \theta+\sqrt{2}=0 $$

In a certain kind of electrical circuit, the current \(I(t)\) measured in amperes at time \(t\) seconds is given by $$ I(t)=10 \cos \left(120 \pi t+\frac{\pi}{3}\right) $$ (a) Change the form of the function \(I(t)\) to the sine function form given in (7). (b) Give the period of the sine function in part (a) and the phase shift. Use the sine function to sketch two cycles of the graph of \(I(t)\).

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \sin \left(\arctan \frac{1}{4}\right) $$

In Problems \(17-20,\) express the given angle in decimal notation. $$ 143^{\circ} 7^{\prime} 2^{\prime \prime} $$

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=3 \pi / 4 $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \tan 195^{\circ} $$

Use the given information to find the values of the remaining five trigonometric functions. $$ \tan x=-2, \pi / 2

Reduce the given expression to a single trigonometric function. $$ \sin t \cos t \tan t \sec t \cot t $$

In Problems \(19-46,\) find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \cos ^{2} x-1=0 $$

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \cos (\arctan (-2)) $$

In Problems \(17-20,\) express the given angle in decimal notation. $$ 5^{\circ} 10^{\prime} $$

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=11 \pi / 6 $$

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \cos 345^{\circ} $$

Use the given information to find the values of the remaining five trigonometric functions. $$ \cot x=\frac{1}{2}, \quad \pi