Exam 9: Trigonometric Identities, Models, and Complex Numbers

Show that $\cos 3 t=4 \cos ^{3} t-3 \cos t$ .

If $s(x)=\sin (2 x)+\sin (x)$ , then $s(x)$ can also be written in the form $s(x)=$ --------- $\sin ($ ---------- $x)(\cos$ ---------- $x)$ .

If $\frac{3 \pi}{2}<\theta<2 \pi$ and $\sin (\theta)=\frac{-8}{9}$ , find $\sin (2 \theta), \cos (2 \theta)$ , and $\tan (2 \theta)$ exactly.

A ferris wheel sitting on the ground is 20 meters in diameter and makes one revolution every 5 minutes. If you start in the 9 o'clock position at $t=0$ and the wheel is rotating clockwise, when is the first time that are you 15 meters above the ground?

A mass attached to a spring moves horizontally on a frictionless track. Its displacement from the rest position at time $t$ is given by $x=0.4 \cos (6 t)$ . What is the furthest distance from the rest position that the mass will achieve? The displacement is measured in meters.

Solve $\csc x=2$ for $0 \leq x \leq 2 \pi$ .

Graph $y=\tan t$ and use that graph to approximate the solution of $0.49=\tan t$ on $0 \leq t \leq \frac{\pi}{2}$ . Give $t$ in degrees to 2 decimal places.

Does $\frac{(\cos \theta-\sin \theta)^{2}-(\cos \theta+\sin \theta)^{2}}{2 \sin 2 \theta}=1$ ?

Find the smallest value of $t$ such that $t>0$ and $\cos (8 t)-\cos (5 t)=0$ .

Find the smallest value of $t$ such that $t>0$ and $\sin (8 t)-\sin (5 t)=0$ .

Select the function that is equivalent to $f(x)=\cos 5 x+\cos 11 x$ .

A ferris wheel sitting on the ground is 20 meters in diameter and makes one revolution every 7 minutes. If you start in the 9 o'clock position $t=0$ and the wheel is rotating counterclockwise, write a formula for your height above the ground at time $t$ .

Graph $y=\sin t$ and use that graph to approximate the solution of $0.65=\sin t$ on $0 \leq t \leq \frac{\pi}{2}$ . Give $t$ in degrees to 2 decimal places.

An acoustic beat produces a combined sound wave represented by the function $f(x)=\cos (2 \pi \cdot 41 x)+\cos (2 \pi \cdot 53 x)$ .

Find all solutions to $2 \cos t=0.14$ for $-2 \pi<t<-\pi$ . Give answers correct to 3 decimal places.

How many solutions does $\cos ^{4} \theta-\sin ^{4} \theta=\frac{1}{3}$ have for $0 \leq \theta \leq 4 \pi$ ?

The following table gives $A(t)$ , the percentage of the electorate favoring candidate $A$ during the 12 months preceding a presidential election. Time, $t$ , is measured in months, and $t=0$ is a year before election day. Assume that $A(t)$ is approximately trigonometric. A second candidate, candidate $B$ , has a percentage of support given by $B(t)=30+15 \sin \left(\frac{\pi}{6} t\right)$ . What is the largest value of $t, 0 \leq t \leq 12$ , at which the two candidates are tied for electoral support? Round to 2 decimal places.

Find the exact value of $\cos 435^{\circ}$ .

Using the sum or difference formulas, $5 \sin t+3 \cos t=$ --------- $\sin (t+$ -----------). Round both answers to 4 decimal places.

The baseball field's usage (in people per week) is seasonal with the peak in mid-July and the low in mid-January. The usage is 2,000 in July and 500 in January. Find a trig function $s=f(t)$ representing the usage at time $t$ months after mid-January.