Chapter 9

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{1}{3 \sqrt{7}}$$

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$\frac{\pi}{12}$$

Determine what fraction of the circumference of the unit circle each value of s represents. For example, \(s=\pi\) represents \(\frac{1}{2}\) of the circumference of the unit circle. Do not use a calculator. $$s=\frac{7 \pi}{6}$$

To show that sec(- \(x\) ) = sec \(x\) for all \(x\) in the domain, we begin by writing $$ \sec (-x)=\frac{1}{\cos (-x)} $$ and then use the fact that \(\cos (-x)=\cos x\) for all \(x\) to complete the argument. Use this method to prove each of the following. $$\tan (-x)=-\tan x$$

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(\frac{\cos x}{\sin x}=\)_____ \(\mathbf{I}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

A note on the piano has frequency \(F\). Suppose the maximum displacement at the center of the piano wire is given by \(s(0)\). Find constants a and \(\omega\) so that the equation \(s(t)=a \cos \omega t\) models this displacement. Graph s in the viewing window \([0,0.05]\) by \([-0.3,0.3]\). $$F=55 ; s(0)=0.14$$

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(a=12\) yards; \(c=37\) yards

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 135^{\circ} & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & & & -\sqrt{2} & \sqrt{2} \\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{1}{\sqrt{x}}$$

What fraction of a complete revolution is each of the following angles? (a) \(180^{\circ}\) (b) \(40^{\circ}\) (c) \(1^{\circ}\)

To show that sec(- \(x\) ) = sec \(x\) for all \(x\) in the domain, we begin by writing $$ \sec (-x)=\frac{1}{\cos (-x)} $$ and then use the fact that \(\cos (-x)=\cos x\) for all \(x\) to complete the argument. Use this method to prove each of the following. $$\cot (-x)=-\cot x$$

A note on the piano has frequency \(F\). Suppose the maximum displacement at the center of the piano wire is given by \(s(0)\). Find constants a and \(\omega\) so that the equation \(s(t)=a \cos \omega t\) models this displacement. Graph s in the viewing window \([0,0.05]\) by \([-0.3,0.3]\). $$F=220 ; s(0)=0.06$$

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(b=15\) feet; \(c=17\) feet

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 150^{\circ} & & -\frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{3} & & & 2\\\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{1}{\sqrt{x-1}}$$

What fraction of a complete revolution is each of the following angles? (a) \(\frac{\pi}{6}\) (b) \(\frac{\pi}{2}\) (c) \(2 \pi\)

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\frac{1}{2} \csc x$$

Write the equation, and then determine the amplitude, period, and frequency of the simple harmonic motion of a particle moving uniformly around a circle of radius 2 units, with angular speed (a) 2 radians per second and (b) 4 radians per second.

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. A=28.0^{\circ} ; c=17.4 \text { feet }

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 210^{\circ} & -\frac{1}{2} & &\frac{\sqrt{3}}{3} & \sqrt{3} & & \- 2\\\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{6}{\sqrt{1-x^{2}}}$$

An angle measures \(x\) degrees. (a) What is the measure of its complement? (b) What is the measure of its supplement?

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=2 \sec \frac{1}{2} x$$

What are the period \(P\) and frequency \(T\) of the oscillation of a pendulum of length \(\frac{1}{2}\) foot? (Hint:\(P=2 \pi \sqrt{\frac{L}{32}},\) where \(L\) is the length of the pendulum in feet and \(P\) is in seconds.)

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. B=46.0^{\circ} ; c=29.7 \text { meters }

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 240^{\circ} &-\frac{\sqrt{3}}{2} & -\frac{1}{2} & & & -2 & \frac{2}{3}\\\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{7}{\sqrt{4-x^{2}}}$$

An angle measures \(x\) radians. (a) What is the measure of its complement? (b) What is the measure of its supplement?

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\tan \frac{1}{2} x$$

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. B=73.0^{\circ} ; b=128 \text { inches }

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\tan 30^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(5,-12)$$

March each function defined in Column I with the appropriate description in Column \(\mathrm{II.}\) Do not use a calculator. I $$y=3 \sin (2 x-4)$$ II A. Amplitude \(=2 ;\) period \(=\frac{\pi}{2} ;\) phase shift \(=\frac{3}{4}\) B. Amplitude \(=3 ;\) period \(=\pi ;\) phase shift \(=2\) C. Amplitude \(=4 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{2}{3}\) D. Amplitude \(=2 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{4}{3}\)

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\cot \left(x+\frac{\pi}{3}\right)$$

The formula for the up-and-down motion of a weight on a spring is given by $$s(t)=a \sin \sqrt{\frac{k}{m}} t$$ If the spring constant \(k\) is \(4,\) what mass \(m\) must be used to produce a period of 1 second?

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cot 30^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-12,-5)$$

March each function defined in Column I with the appropriate description in Column \(\mathrm{II.}\) Do not use a calculator. I $$y=2 \sin (3 x-4)$$ II A. Amplitude \(=2 ;\) period \(=\frac{\pi}{2} ;\) phase shift \(=\frac{3}{4}\) B. Amplitude \(=3 ;\) period \(=\pi ;\) phase shift \(=2\) C. Amplitude \(=4 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{2}{3}\) D. Amplitude \(=2 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{4}{3}\)

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=2 \csc \frac{1}{2} x$$

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(a=76.4\) yards; \(b=39.3\) yards

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\sin 30^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-3,4)$$

Find the degree measure of the smaller angle formed by the hands of a clock at the following times. Do not use a calculator. $$3: 15$$

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=3 \csc 2 x$$

The position of a weight attached to a spring is $$s(t)=-5 \cos 4 \pi t$$ inches after \(t\) seconds. (a) What is the maximum height to which the weight rises above the equilibrium position? (b) What are the frequency and period? (c) When does the weight first reach its maximum height? (d) Calculate and interpret \(s(1.3)\)

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(a=958\) meters; \(b=489\) meters

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cos 30^{\circ}$$

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-4,-3)$$

Find the degree measure of the smaller angle formed by the hands of a clock at the following times. Do not use a calculator. $$9: 00$$

March each function defined in Column I with the appropriate description in Column \(\mathrm{II.}\) Do not use a calculator. I $$y=2 \sin (4 x-3)$$ II A. Amplitude \(=2 ;\) period \(=\frac{\pi}{2} ;\) phase shift \(=\frac{3}{4}\) B. Amplitude \(=3 ;\) period \(=\pi ;\) phase shift \(=2\) C. Amplitude \(=4 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{2}{3}\) D. Amplitude \(=2 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{4}{3}\)