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Problem 1

Evaluate the following expressions:

(a) $\tan(45^\circ)$

(b) $\cos\left(\frac {7\pi}{4}\right)$

(c) $\sin\left(\frac {5\pi}{3}\right)$

(d) $\csc(135^\circ)$

(e) $\cot(945^\circ)$

(f) $\sin(\pi \sin(\pi/6))$

(g) $\tan(21\pi)$

(h) $\sec( - 585^\circ)$

Problem 2

Using the unit circle, find $\sin \left( x + \frac {\pi}{2} \right)$ and $\cos \left( x + \frac {\pi}{2} \right)$ in terms of $\sin x$ and $\cos x$ .

Problem 3

In triangle $ABC$ , $\angle B = 90^\circ$ , $\sin A = 7/9$ , and $BC = 21$ . What is $AB$ ?

Problem 4

What does the graph of $\sin 4x$ look like compared to the graphs of $\sin x$ and $\cos x$ ? What about the graph of $2\sin \left( 3x + \frac {\pi}{4} \right) - 1$ ?

Problem 5

Find the value of $\tan(\pi/12) \cdot \tan(2\pi/12) \cdot \tan(3\pi/12) \cdots \tan(5\pi/12).$

Problem 6

Suppose that parallelogram $ABCD$ has $\angle A = \angle C = 30^\circ$ , $\angle B = \angle D = 150^\circ$ , and the shorter diagonal $BD$ has length 2. If the height of the parallelogram is $a$ , find the perimeter of $ABCD$ in terms of $a$ .

Problem 7

Given a positive number $n$ and a number $c$ satisfying $- 1 < c < 1$ , for how many values of $q$ with $0 \leq q < 2\pi$ is $\sin nq = c$ ? What if $c = 1$ or $c = - 1$ ?