Chapter 9

Refer to the equations in the definition of simple harmonic motion in this section. and consider the following equation. \(s(t)=5 \cos 2 t, \quad\) where \(t\) is time in seconds What is the amplitude of this motion?

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. $$s=\frac{\pi}{3}$$

Fill in the blanks with the appropriate short answers. Do not use a calculator. An angle of \(360^{\circ}\) has an equivalent radian measure of ___.

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. $$s=\frac{2 \pi}{3}$$

Fill in the blanks with the appropriate short answers. Do not use a calculator. An angle of \(\pi\) radians has an equivalent degree measure of ___.

Refer to the equations in the definition of simple harmonic motion in this section. and consider the following equation. \(s(t)=5 \cos 2 t, \quad\) where \(t\) is time in seconds What is the frequency?

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. $$s=\frac{\pi}{4}$$

Fill in the blanks with the appropriate short answers. Do not use a calculator. The least positive angle coterminal with \(-180^{\circ}\) has degree measure___.

Refer to the equations in the definition of simple harmonic motion in this section. and consider the following equation. \(s(t)=5 \cos 2 t, \quad\) where \(t\) is time in seconds What is \(s(0) ?\)

How would the six trigonometric functions of \(90^{\circ}\) and \(-270^{\circ}\) compare? Why?

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. $$s=\frac{3 \pi}{4}$$

Fill in the blanks with the appropriate short answers. Do not use a calculator. The complement of a \(40^{\circ}\) angle is ___ and the supplement of a \(40^{\circ}\) angle is___.

Refer to the equations in the definition of simple harmonic motion in this section. and consider the following equation. \(s(t)=5 \cos 2 t, \quad\) where \(t\) is time in seconds \text { What is } s\left(\frac{\pi}{2}\right) ?

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

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. $$s=\frac{\pi}{6}$$

Refer to the equations in the definition of simple harmonic motion in this section. and consider the following equation. \(s(t)=5 \cos 2 t, \quad\) where \(t\) is time in seconds What is the range of this function?

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

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. $$s=\frac{5 \pi}{6}$$

Fill in the blanks with the appropriate short answers. Do not use a calculator. A formula for \(v\) relating \(v, \omega,\) and \(r\) is___.

Tell whether each statement is true or false. If false, tell why. The least positive number \(k\) for which \(x=k\) is an asymptote for the tangent function is \(\frac{\pi}{2}\).

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{3}{\sqrt{13}}$$

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. $$s=3 \pi$$

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$30^{\circ}$$

Tell whether each statement is true or false. If false, tell why. The least positive number \(k\) for which \(x=k\) is an asymptote for the cotangent function is \(\frac{\pi}{2}\)

Suppose that a weight on a spring has initial position s(0) and period \(P .\) Do not use a calculator. (a) Find a finction \(s\) given by \(s(t)=a\) cos \(\omega t\) that models the displacement of the weight. (b) Evaluate \(s(1)\). Is the weight moving upward, downward, or neither when \(t=1 ?\) \(s(0)=5\) inches; \(P=1.5\) seconds

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{5}{\sqrt{29}}$$

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. $$s=\frac{5 \pi}{2}$$

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$60^{\circ}$$

Tell whether each statement is true or false. If false, tell why. The secant and cosecant functions are undefined for the same values.

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-0.5)$$

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 30^{\circ} & \frac{1}{2} & \frac{\sqrt{3}}{2} & & & \frac{2 \sqrt{3}}{3} & 2 \\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{3}{2 \sqrt{10}}$$

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. $$s=\frac{3 \pi}{2}$$

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$45^{\circ}$$

Tell whether each statement is true or false. If false, tell why. The tangent and secant functions are undefined for the same values.

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-2.5)$$

Suppose that a weight on a spring has initial position s(0) and period \(P .\) Do not use a calculator. (a) Find a finction \(s\) given by \(s(t)=a\) cos \(\omega t\) that models the displacement of the weight. (b) Evaluate \(s(1)\). Is the weight moving upward, downward, or neither when \(t=1 ?\) \(s(0)=-4\) inches; \(P=1.2\) seconds

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 45^{\circ} & & & 1 & 1 & & \\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{4}{5 \sqrt{41}}$$

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$\frac{\pi}{3}$$

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. $$\sec (-x)=\sec x$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\tan \left(-\frac{\pi}{7}\right)$$

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 60^{\circ} & & \frac{1}{2} & \sqrt{3} & & 2 & \\ \hline \end{array}$$

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{1}{2 \sqrt{5}}$$

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$\frac{\pi}{4}$$

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. $$s=\frac{5 \pi}{4}$$

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. $$\csc (-x)=-\csc x$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\cot \left(-\frac{4 \pi}{7}\right)$$

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 \([-0.3,0.3]\). $$F=110 ; s(0)=0.11$$

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 120^{\circ} & \frac{\sqrt{3}}{2} & & -\sqrt{3} & & & \frac{2 \sqrt{3}}{3} \\ \hline \end{array}$$