Exam 9: Trigonometric Identities, Models, and Complex Numbers

If $-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$ , in what quadrant is $\sin t=-0.64$ ?

Either show the following equation is true, or find a value of $x$ for which the equation is false: $\sin (4 x)+\sin (2 x)=\sin (6 x)$

Does $\sin \frac{1}{2} t=\sqrt{\frac{1-\cos t}{2}}$ ?

If $0 \leq t \leq \pi$ , in what quadrant is $\cos t=0.31$ ?

The population in a town oscillates over a 13 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.

How many solutions to $\cos x=\frac{1}{4}$ are there for $0 \leq x \leq \pi$ ?

Find all solutions to the equation $7 \sin (x-1)=2$ for $0 \leq x \leq 2 \pi$ . Give your answers to 3 decimal places.

Find the smallest value of $t$ such that $t>0$ and $\sin (8 t)+\sin (7 t)=0$ .

Solve $\tan x=3$ for $0 \leq x \leq 2 \pi$ .

What is the smallest positive solution to $5 \sin ^{2} x-\cos ^{2} x=1$ ? Round to 2 decimal places.

A ferris wheel sitting on the ground is 26 meters in diameter and makes one revolution every 3 minutes. If you start in the 9 o'clock position at $t=0$ and the wheel is rotating counterclockwise, when is the first time that are you 6.5 meters above the ground?

Solve $\tan x-2 \sin x=0$ for $0 \leq x \leq 2 \pi$ .

Find a formula for a deer population which oscillates over a 6 year period between a low of 1,000 in year $t=0$ and a high of 2,700 in year $t=3$ .

Find the smallest value of $t$ such that $t>0$ and $\cos (6 t)+\cos (3 t)=0$ .

Which of the following are defined? A) $\sin ^{-1} 3.45$ B) $\cos ^{-1} 3.45$ C) $\tan ^{-1} 3.45$ D) $\sec ^{-1} 3.45$ E) $\csc ^{-1} 3.45$ F) $\cot ^{-1} 3.45$

Find all values of $\theta$ such that $\tan \theta=3.161$ and $360^{\circ} \leq \theta \leq 720^{\circ}$ . Give answers correct to 3 decimal places.

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. If $A(t)$ were approximately trigonometric, its formula could be written $A(t)=----------\cos (\ldots \pi t)+\ldots$ . Round the second answer to 3 decimal places.

If $\pi<\theta<\frac{3 \pi}{2}$ and $\cos (\theta)=\frac{4}{11}$ , find $\cos (2 \theta), \sin (2 \theta)$ , and $\tan (2 \theta)$ exactly.

At which of the following values of $x$ do the graphs of $y=25+\frac{1}{2} t$ and $y=25+\frac{1}{2} t-2 \cos \left(\frac{t}{2}\right)$ intersect?

A weight attached to a spring starts at rest. Let $d$ be the displacement, in centimeters, from the weight's rest position. We raise the spring $21 \mathrm{~cm}$ above its rest position and release it at time $t=0$ , where $t$ is in seconds. The weight bobs up and down once every second for a few seconds. The amplitude of the spring's motion shrinks $b$ half every second. Give a model for the spring's motion.