Trigonometric Functions: Mixed Practice

Before you begin#

Four­teen ques­tions from across the whole chap­ter. Two habits will carry you through. When eval­u­at­ing, bring every angle back to the first quad­rant and set­tle the sign from the quad­rant it came from. When prov­ing, start from the more com­pli­cated side and sim­plify it; that is almost always eas­ier than build­ing up the sim­ple side.

Ques­tions#

1. Con­vert $-37^\circ 30'$ to radi­ans and $\displaystyle \tfrac{11\pi}{18}$ to degrees.
2. A rail­way track curves along an arc of radius $1.5$ km. Through what angle (in degrees) does a train turn while it trav­els $500$ m? Take $\displaystyle \pi = \tfrac{22}{7}$.
3. Given $\displaystyle \sec x = -\tfrac{5}{3}$ with $x$ in III, find the other five func­tions.
4. Eval­u­ate: $\sin 765^\circ$; $\displaystyle \cot(-\tfrac{15\pi}{4})$; $\displaystyle \cos \tfrac{-19\pi}{3}$.
5. Find: $\displaystyle \sin^2 \tfrac{\pi}{6} + \cos^2 \tfrac{\pi}{3} - \tan^2 \tfrac{\pi}{4}$.
6. Prove: $\displaystyle 2\sin^2 \tfrac{3\pi}{4} + 2\cos^2 \tfrac{\pi}{4} + 2\sec^2 \tfrac{\pi}{3} = 10$.
7. Find $\cos(x - y)$ given $\displaystyle \cos x = \tfrac{4}{5}$, $\displaystyle \sin y = \tfrac{5}{13}$, $x, y$ acute.
8. Prove: $\sin(n + 1)x \sin(n + 2)x + \cos(n + 1)x \cos(n + 2)x = \cos x$.
9. Prove: $\cos^2 2x - \cos^2 6x = \sin 4x \sin 8x$.
10. Prove: $\displaystyle \frac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x$.
11. Prove: $\cot 4x(\sin 5x + \sin 3x) = \cot x(\sin 5x - \sin 3x)$.
12. Given $\displaystyle \sin x = \tfrac{1}{4}$ with $x$ in II, find $\sin 2x$ and $\displaystyle \cos \tfrac{x}{2}$.
13. Prove: $\tan 3x \tan 2x \tan x = \tan 3x - \tan 2x - \tan x$.
14. Prove: $\cos 6x = 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1$.

Answers#