Exam 9: Trigonometric Identities, Models, and Complex Numbers

Write $\frac{3 \sin (x-3)}{4 \cos (x-3)}$ in terms of the tangent function.

Write $-8 \sin (3 t)+11 \cos (3 t)$ in the form $A \sin (B t+C)$ . Round all numbers to 3 decimal places if necessary.

The squirrel population in a region varies periodically depending on the time of year. If the starting population is 4,000 squirrels and the population fluctuations up and down by 1,500 squirrels on an annual cycle (more in summer and less in winter). Due to conservation and feeding efforts, the average number of squirrels increases by 100 per year. Give a model for the number of squirrels.

The population in a town oscillates over a 17 year period beginning with a high of 3,000 people in year $t=0$ and a low of 2,300. Find a formula for the town's population.

A ferris wheel sitting on the ground is 24 meters in diameter and makes one revolution every 5 minutes. If you start in the 2 o'clock position $t=0$ and the wheel is rotating clockwise, write a formula for your height above the ground at time $t$ .

Does $\cos (3 t)=3 \cos ^{3} t-3 \cos t$ ?

If $0 \leq t \leq \pi$ , in what quadrant is $\cos t=-0.69$ ?

How many solutions to $\sin x=\frac{-1}{3}$ are there for $0 \leq x \leq \pi$ ?

Find all solutions to $\sin \theta=0.8$ on the interval $3 \pi \leq \theta \leq 5 \pi$ . Give your answers correct to 3 decimal places.

Select the function that is equivalent to $f(x)=\sin 7 x-\sin 11 x$ .

Does $\cos \left(-75^{\circ}\right)=\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}+\frac{1}{2} \cdot \frac{\sqrt{2}}{2} ?$

Write $8 \sin (4 t)-9 \cos (4 t)$ in the form $A \sin (B t+C)$ . Round all numbers to 3 decimal places if necessary.

Two weights (weight 1 and weight 2 ) are suspended from the ceiling by springs. At time $\mathrm{t}=0$ ( $\mathrm{t}$ in seconds), the weights are set in motion and begin bobbing up and down. Eventually, however, the oscillation of both weights dies down. The following equations describe the distance of each weight from the ceiling as a function of time: $d_{1}=6+5 \cos (\pi t) e^{-0.1 t} \text { and } d_{2}=5+4 \cos (2 \pi t) e^{-0.3 t}$ At what time are the two weights furthest apart?

How many solutions to $\sin x=\frac{1}{6}$ are there for $0 \leq x \leq \pi$ ?

Find the smallest value of $t$ such that $t>0$ and $\cos (10 t)-\cos (5 t)=0$ .

Which of the following statements are identities?

Find the smallest value of $t$ such that $t>0$ and $\cos (10 t)+\cos (9 t)=0$ .

Calculate $\tan 15^{\circ}$ exactly.

What is the smallest positive solution to $\cos ^{4} \theta-\sin ^{4} \theta=\frac{1}{4}$ ? Round your answer to 2 decimal places.

Write $\frac{5 \sin (x-5)}{8 \cos (x-5)}$ in terms of the tangent function.