Trigonometric functions: practice problems

Radians and general angles

$180^\circ = \pi$ radians, so $30^\circ = \frac{\pi}{6}$, $45^\circ = \frac{\pi}{4}$, $60^\circ = \frac{\pi}{3}$ and $90^\circ = \frac{\pi}{2}$.

If the terminal side of $\theta$ meets the unit circle at $(x,\ y)$, then $\cos\theta = x$, $\sin\theta = y$ and $\tan\theta = \frac{y}{x}$. Angles differing by $2n\pi$ have the same values.

Periods and translations of graphs

The graph of $y = a\sin(b\theta - c) + d$ ($a > 0$, $b > 0$) is the graph of $y = a\sin b\theta$ shifted by $\frac{c}{b}$ in the $\theta$-direction and by $d$ vertically.

• Period: $\frac{2\pi}{b}$ ($\frac{\pi}{b}$ for $\tan$)
• Maximum $a + d$, minimum $-a + d$

Trigonometric equations and inequalities

To solve $\sin\theta = \frac{1}{2}$ on $0 \leq \theta < 2\pi$, find the points of the unit circle with $y$-coordinate $\frac{1}{2}$: $\theta = \frac{\pi}{6},\ \frac{5\pi}{6}$. For $\sin\theta > \frac{1}{2}$, take the arc between them: $\frac{\pi}{6} < \theta < \frac{5\pi}{6}$.

For $\sin\left(\theta + \frac{\pi}{3}\right) = \frac{1}{2}$ or $\cos 2\theta = \frac{1}{2}$, substitute $t = \theta + \frac{\pi}{3}$ or $t = 2\theta$ and find the range of $t$ first.

For quadratics such as $2\cos^2\theta + 3\sin\theta - 3 = 0$, use $\cos^2\theta = 1 - \sin^2\theta$ to get an equation in $\sin\theta$, remembering $-1 \leq \sin\theta \leq 1$.