Exam 8: The Unit Circle and Functions of Trigonometry

A central angle of a circle with radius 35 centimeters cuts off an arc of 100 centimeters. Find each measure. (a) The radian measure of the angle. (b) The area of the sector with this central angle.

If $(-3,-5)$ is on the terminal side of an angle $\theta$ in standard position, find $\sin \theta, \cos \theta$ , and $\tan \theta$ .

Use a calculator to approximate $\theta$ to the nearest tenth in the interval $\left[0,90^{\circ}\right]$ , if $\cos \theta=0.9086146585$ .

graph the given function over a two-period interval. Identify asymptotes when applicable. - $y=3 \cos (x-\pi)+2$

To find the height of a tree, an arborist found that the angle of elevation from a point 52.6 feet from the base of the tree is $43^{\circ} 21^{\prime}$ . What is the height of the tree?

If $\cos \theta=\frac{5}{13}$ and $\theta$ is in quadrant IV find the values of the other trigonometric functions of $\theta$ .

the formula $s(t)=-5 \cos 3 \pi t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds. -Find the maximum height that the weight rises above the equilibrium position of $y=0$ .

Find the exact values of each part labeled with a letter.

graph the given function over a two-period interval. Identify asymptotes when applicable - $y=2 \cos (x+\pi)+1$

graph the given function over a two-period interval. Identify asymptotes when applicable. - $y=-\sin 4 x$

A wheel is turning at a speed of 72 revolutions per minute. Through how many degrees does a point on the edge of the wheel move in 3 seconds?

graph the given function over a two-period interval. Identify asymptotes when applicable. -The average monthly low temperature (in $\mathrm{F}^{\circ}$ ) in London can be modeled using the trigonometric function defined by $f(x)=10 \sin \left[\frac{\pi}{6}(x-4.2)\right]+46$ , where $x$ is the month and $x=1$ corresponds to January. (a) Graph $f$ over the interval $1 \leq x \leq 25$ . (b) Determine the amplitude, period, phase shift, and vertical translation of $f$ . (c) What is the average monthly low temperature for the month of October? (d) Find the maximum and minimum average monthly low temperatures and the months when they occur. (e) What would be an approximation for the average yearly low temperature in London? How is this related to the vertical translation of the sine function in the formula for $f$ ?

A jet leaves an airport on a bearing of $\mathrm{N} 59^{\circ} \mathrm{E}$ and travels for 389 miles. It then turns and continues on a bearing of $\mathrm{S} 31^{\circ} \mathrm{E}$ for 207 miles. How far is the jet from the airport?

the formula $s(t)=-4 \cos 3 t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds -When does the weight reach its maximum height if $t \geq 0$ ?

A wheel is turning at a speed of 100 revolutions per minute. Through how many degrees does a point on the edge of the wheel move in 1.05 second?

If $\csc \theta<0$ and $\tan \theta<0$ , in which quadrant does $\theta$ lie?

Use a calculator to approximate the following. (a) $\csc 85^{\circ} 14^{\prime}$ (b) $\cos 101.655^{\circ}$ (c) $\tan 221.125^{\circ}$

graph the given function over a two-period interval. Identify asymptotes when applicable. - $y=\tan \left(x+\frac{\pi}{2}\right)$

Find the exact values of each part labeled with a letter.

A central angle of a circle with radius 50 centimeters cuts off an arc of 120 centimeters. Find each measure. (a) The radian measure of the angle. (b) The area of the sector with this central angle.