Chapter 14

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan (-3 \theta) $$

Graph each function in the interval from 0 to 2\(\pi\). \(y=3 \sin 2\left(x+\frac{\pi}{2}\right)-5\)

Find the complete solution of \(\sin ^{2} \theta+2 \sin \theta+1=0 .\) (Hint: How would you solve \(x^{2}+2 x+1=0 ? )\)

A set of values has a mean of 36 and a standard deviation of \(5 .\) Find the \(z\) -score of the value \(43 .\)

\(\begin{array}{ll}{\text { How can you express } \csc ^{2} \theta-2 \cot ^{2} \theta \text { in terms of } \sin \theta \text { and } \cos \theta ?} \\\ {\mathrm{F} \cdot \frac{1-2 \cos ^{2} \theta}{\sin ^{2} \theta}} & {\text { G. } \frac{1-2 \sin ^{2} \theta}{\sin ^{2} \theta}} \\ {\mathrm{H} \cdot \sin ^{2} \theta-2 \cos ^{2} \theta} & {\text { J. } \frac{1}{\sin ^{2} \theta}-\frac{2}{\tan ^{2} \theta}}\end{array}\)

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan \frac{2 \pi}{5} \theta $$

In \(\triangle X Y Z, m \angle Z=51^{\circ}, x=13 \mathrm{cm},\) and \(y=17 \mathrm{cm} .\) Find \(z\)

Graph each function in the interval from 0 to 2\(\pi\). \(y=\cos (x+\pi)-3\)

a. Open-Ended Write three trigonometric equations whose complete solution is \(\pi+2 \pi n .\) b. Describe how you found the equations in part (a).

Which expression is equivalent to \(\frac{\tan \theta}{\cos \theta-\sec \theta} ?\) \(\begin{array}{llll}{\text { A. } \csc \theta} & {\text { B. } \sec \theta} & {\text { C. }-\csc \theta} & {\text { D. } \tan ^{2} \theta}\end{array}\)

Graph each function in the interval from 0 to 2\(\pi\). \(y=-2 \cos \left(x-\frac{\pi}{3}\right)-4\)

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=2 \sin \theta $$

Show that \((\sec \theta+1)(\sec \theta-1)=\tan ^{2} \theta\) is an identity.

In \(\triangle A B C, m \angle A=87^{\circ}, b=22 \mathrm{m},\) and \(c=19 \mathrm{m} .\) Find \(a\)

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=\cos 2 \theta $$

Show that \(\frac{\cos x}{1-\sin ^{2} x}=\sec x\) is an identity.

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 80^{\circ} $$

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=3 \sin (\theta+2) $$

Graph each function in the interval from 0 to 2\(\pi\). $$ y=\csc (-\theta) $$

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ -50^{\circ} $$

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=-4 \cos 2 \pi \theta $$

Graph each function in the interval from 0 to 2\(\pi\). $$ y=-\cot \theta $$

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ -15^{\circ} $$

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=\cos \theta+1 $$

Graph each function in the interval from 0 to 2\(\pi\). $$ y=-\sec 0.5 \theta $$

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 70^{\circ} $$

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=2 \cos \pi \theta+1 $$

Graph each function in the interval from 0 to 2\(\pi\). $$ y=-\sec (0.5 \theta+2) $$

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 190^{\circ} $$

Parks A segment of a circle is the region formed by an arc of a circle and the line segment joining the endpoints of the arc. (The measure of the arc must be between 0 and 2\(\pi .\) The expression below gives the area \(K\) in square feet of a segment in terms of the radius of the circle \(r\) in feet and the radian measure \(\theta\) of the arc. $$K=\frac{r^{2}}{2}(\theta-\sin \theta)$$ a. Suppose a circular park has a radius of 50 ft. Write an equation for the area of an segment of the park using \(r=50 .\) b. Use trial and error to approximate the measure of \(\theta\) that makes the area of the segment 1500 \(\mathrm{ft}^{2} .\)

Graph each function in the interval from 0 to 2\(\pi\). $$ y=\cot \frac{\theta}{5} $$

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 200^{\circ} $$

Tides One day the tides at a point in Maine could be modeled by \(h=5 \cos \frac{2 \pi}{13} t\) where \(h\) is the height of the tide in feet above the mean water level and \(t\) is the number of hours past midnight. a. At what times that day will the tide be 3 \(\mathrm{ft}\) above the mean water level? b. At what times that day will the tide be at least 3 ft above the mean water level?

Graph each function in the interval from 0 to 2\(\pi\). $$ y=\pi \sec \theta $$

A set of data with a mean of 39 and a standard deviation of 6.2 is normally distributed. Find each value, given its distance from the mean. \(+1\) standard deviation

Which of the following is NOT equal to \(60^{\circ} ?\) $$\begin{array}{llll}{\text { A. } \sin ^{-1} \frac{\sqrt{3}}{2}} & {\text { B. } \cos ^{-1} \frac{1}{2}} & {\text { C. } \tan ^{-1} \sqrt{3}} & {\text { D. } \tan ^{-1} \frac{\sqrt{3}}{3}}\end{array}$$

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 395^{\circ} $$

A set of data with a mean of 39 and a standard deviation of 6.2 is normally distributed. Find each value, given its distance from the mean. \(-2\) standard deviations

In which quadrants are the solutions to \(\tan \theta+1=0 ?\) F. Quadrants I and II H. Quadrants II and IV G. Quadrants II and III J. Quadrants III and IV

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 405^{\circ} $$

A set of data with a mean of 39 and a standard deviation of 6.2 is normally distributed. Find each value, given its distance from the mean. \(+3\) standard deviations

Which of these angles have a sine of about \(-0.6 ?\) $$\begin{array}{llll} \text { I. } {1.143 .1^{\circ}} & {\text { II. } 216.9^{\circ}} & {\text { III. } 323.1^{\circ}}\end{array}$$ A. I and II only B. I and III only C. II and III only D. I, II, and III

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -225^{\circ} $$

A set of data with a mean of 39 and a standard deviation of 6.2 is normally distributed. Find each value, given its distance from the mean. \(-1\) standard deviation

What are the solutions of \(2 \sin \theta-\sqrt{3}=0\) for \(0 \leq \theta<2 \pi ?\) F. \(\frac{\pi}{6}\) and \(\frac{5 \pi}{6}\) H. \(\frac{2 \pi}{3}\) and \(\frac{4 \pi}{3}\) G. \(\frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) J. \(\frac{4 \pi}{3}\) and \(\frac{5 \pi}{3}\)

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -149^{\circ} $$

Suppose \(a>0 .\) Under what conditions for a and \(b\) will \(a \sin \theta=b\) have exactly two solutions in the interval \(0 \leq \theta<2 \pi ?\) A. \(a=b\) C. \(a=-b\) B. \(b>a\) D. \(a>b>-a\)

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 627^{\circ} $$

Solve \(2 \cos \theta=\sqrt{2}\) for \(0 \leq \theta<2 \pi\)

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -281^{\circ} $$