Work each problem.

Find the radius of a circle in which a central angle of $\dfrac{\pi}{6}$ radian determines a sector of area $64 \mathrm{~m}^2$.

The formula $\omega=\dfrac{\theta}{t}$ can be rewritten as $\theta=\omega t$. Using $\omega t$ for $\theta$ changes $s=r \theta$ to $s=r \omega t$. Use the formula $s=r \omega t$ to find the value of the missing variable.

$r=9$ yd, $\omega=\dfrac{2 \pi}{5}$ radians per sec, $t=12$ sec

Find the area of a sector of a circle having radius $r$ and central angle $\theta$. See the example in this section.

$$r=90.0 \mathrm{~km}, \theta=270^{\circ}$$

Graph each function over the interval $[-2 \pi, 2 \pi]$. Give the amplitude. See the example in this section.

$$y=\dfrac{3}{4} \cos x$$

Graph each function over a one-period interval. See the example in this section.

$$y=-2 \sec \dfrac{1}{2} x$$

Convert each degree measure to radians. Leave answers as multiples of $\pi$. See the example in this section.

$$-210^{\circ}$$

Find the exact circular function value for each of the following. See the example in this section.

$$\cos \dfrac{5 \pi}{3}$$

Graph each function over a two-period interval.

$$y=-5 \sin \dfrac{x}{3}$$

A note on the piano has given 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 $[0, .05]$ by $[-.3, .3]$.

$$F=220 ; s(0)=.06$$

The term grade has several different meanings in construction work. Some engineers use the term grade to represent $\dfrac{1}{100}$ of a right angle and express grade as a percent. For instance, an angle of $.9^{\circ}$ would be referred to as a $1 \%$ grade.

In a rapid-transit rail system, the maximum grade allowed between two stations is $3.5 \%$. Express this angle in degrees and radians.

A central angle of a circle with radius $150$ cm cuts off an arc of $200$ cm. Find each measure.

the area of a sector with that central angle

A central angle of a circle with radius $150$ cm cuts off an arc of $200$ cm. Find each measure.

the radian measure of the angle

The term grade has several different meanings in construction work. Some engineers use the term grade to represent $\dfrac{1}{100}$ of a right angle and express grade as a percent. For instance, an angle of $.9^{\circ}$ would be referred to as a $1 \%$ grade.

By what number should you multiply a grade to convert it to radians?

The formula $\omega=\dfrac{\theta}{t}$ can be rewritten as $\theta=\omega t$. Using $\omega t$ for $\theta$ changes $s=r \theta$ to $s=r \omega t$. Use the formula $s=r \omega t$ to find the value of the missing variable.

$r=6$ cm, $\omega=\dfrac{ \pi}{3}$ radians per sec, $t=9$ sec