Exam 9: Trigonometric Identities, Models, and Complex Numbers

The deer population in a state park is modelled by $f(t)=55 \sin \left(\frac{\pi t}{6}\right)+220$ where $t$ is the number of months since January 1, 2005. Evaluate $f(6)-f(3)$ and interpret the result. Round to the nearest whole number.

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

Using the sum or difference formulas, $2 \sin 3 t-4 \cos 3 t=$ ------------- $\sin ( ---------- t+$ ------------). Round all answers to 4 decimal places if necessary.

Write $\frac{3 \cos (x-6)}{8 \sin (x-6)}$ in terms of the cotangent function.

Which formula is the sinusoidal form $f(t)=\sin t+\cos t$ .

How many solutions does $6 \sin ^{2} x-\cos ^{2} x=3$ have for $0 \leq x \leq 2 \pi$ ?

Find the smallest value of $t$ such that $t>0$ and $\sin (4 t)+\sin (5 t)=0$ .

Select the function that is equivalent to $f(x)=\sin 7 x+\sin 17 x$ .

Two weights (weight 1 and weight 2 ) are suspended from the ceiling by springs. At time $t=0$ ( $t$ 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: $d_{1}=5+4 \cos (\pi t) e^{-0.3 t} \text { and } d_{2}=3+3 \cos (2 \pi t) e^{-0.5 t}$ Which weight has oscillations which die down the slowest?

The deer population in a state park is modelled by $f(t)=50 \sin \left(\frac{\pi t}{6}\right)+230$ where $t$ is the number of months since January 1,2005 . If $0 \leq t \leq 12$ , find the value(s) of $t$ at which the deer population is equal to 272 . Round your answer to the nearest tenth.

Find all solutions (if possible) with $t$ in radians: $2 \cos (t-1)=4$ . Give answers correct to 3 decimal places.

How many solutions does $4 \sin ^{2} x-\cos ^{2} x=4$ have for $0 \leq x \leq 2 \pi$ ?

Find all solutions (if possible) for $t$ in radians: $4 \sin t=3$ . Give answers correct to 3 decimal places.

Write $g(t)=8 \sin 3 t)+15 \cos (3 t)$ in the form $A \sin (B t+h)$ . Round numbers to 3 decimal places when necessary.

What is the smallest positive solution to $\cos ^{3} x+\cos x \sin ^{2} x=4 \sin x$ ? Round to 2 decimal places.

Using the sum or difference formulas, $3 \sin 2 t-4 \cos 2 t=\ldots \sin (\ldots t+\ldots)$ . Round all answers to 4 decimal places if necessary.

A mass attached to a spring moves horizontally on a frictionless track. Its velocity at time $t$ is given by $x=-5.6 \sin (8 t)$ . What is the maximum velocity that the mass will achieve? The displacement is measured in meters and the time is measured in seconds.

If $s(x)=\sin (3 x)+\sin (4 x)$ , then $s(x)$ can also be written in the form $s(x)=$ ---------- $\sin ($ -------------- $x)(\cos$ ---------- $x)$ .

Using the sum or difference formulas, $3 \sin t+2 \cos t=\ldots \sin (t+\ldots)$ . Round both answers to 4 decimal places.

Either show the following equation is true, or find a value of $x$ for which the equation is false: $\sin (8)=2 \sin (4) \cos (4)$