Trigonometric values of any angle

With the allied angle rule you can now find the exact value of a trigono­met­ric ratio for any angle built from $30°$, $45°$ or $60°$, how­ever large, and for angles given in radi­ans. This les­son shows the rou­tine: remove whole turns, write the angle as $180° \pm \theta$ or $360° - \theta$ (or $90°$, $270°$ forms), apply the rule, and read the value from the table. It also cov­ers the angles on the axes, where the ratios are $0$, $1$ or $-1$, and the angles at which $\sin$, $\cos$ and $\tan$ van­ish.

Angles between $90°$ and $360°$#

Write the angle as $180° - \theta$, $180° + \theta$ or $360° - \theta$ with $\theta$ acute. These forms keep the ratio, so only the sign has to be decided.

Exam­ple 1#

Find $\sin 120°$, $\cos 120°$, $\cos 240°$ and $\tan 210°$.

$$\displaystyle \sin 120° = \sin(180° - 60°) = +\sin 60° = \frac{\sqrt{3}}{2}$$

since $120°$ is in the sec­ond quad­rant, where $\sin$ is pos­i­tive.

$$\displaystyle \cos 120° = \cos(180° - 60°) = -\cos 60° = -\frac{1}{2}$$

$$\displaystyle \cos 240° = \cos(180° + 60°) = -\cos 60° = -\frac{1}{2}$$

$$\displaystyle \tan 210° = \tan(180° + 30°) = +\tan 30° = \frac{1}{\sqrt{3}}$$

since $\tan$ is pos­i­tive in the third quad­rant.

Exam­ple 2#

Find $\sec 330°$, $\csc 240°$, $\tan 135°$ and $\cos 225°$.

$$\displaystyle \sec 330° = \sec(360° - 30°) = +\sec 30° = \frac{2}{\sqrt{3}}$$

$$\displaystyle \csc 240° = \csc(180° + 60°) = -\csc 60° = -\frac{2}{\sqrt{3}}$$

$$\tan 135° = \tan(180° - 45°) = -\tan 45° = -1$$

$$\displaystyle \cos 225° = \cos(180° + 45°) = -\cos 45° = -\frac{1}{\sqrt{2}}$$

A point to watch: $\displaystyle \cos 30° = \frac{\sqrt{3}}{2}$ and $\displaystyle \sin 60° = \frac{\sqrt{3}}{2}$ are equal, but $\displaystyle \cos 60° = \frac{1}{2}$. Check which ref­er­ence angle you are using before you read the table.

Angles larger than $360°$#

First take away as many com­plete turns ($360°$, $720°$, $1080°$, $1440°$, ...) as you can. Whole turns change noth­ing.

Exam­ple 3#

Find $\sin 750°$ and $\tan 1470°$.

$750° = 720° + 30° = 2 \times 360° + 30°$, so

$$\displaystyle \sin 750° = \sin 30° = \frac{1}{2}$$

$1470° = 1440° + 30° = 4 \times 360° + 30°$, so

$$\displaystyle \tan 1470° = \tan 30° = \frac{1}{\sqrt{3}}$$

Angles in radi­ans#

Either change to degrees (replace $\pi$ by $180°$), or remove mul­ti­ples of $2\pi$ directly.

Exam­ple 4#

Find $\displaystyle \cos\frac{33\pi}{4}$.

In degrees. $\displaystyle \frac{33\pi}{4} = 33 \times 45° = 1485°$, and $1485° = 1440° + 45°$. So

$$\displaystyle \cos\frac{33\pi}{4} = \cos 1485° = \cos 45° = \frac{1}{\sqrt{2}}$$

In radi­ans. $\displaystyle \frac{33\pi}{4} = 8\pi + \frac{\pi}{4}$, and $8\pi$ is four whole turns. So $\displaystyle \cos\frac{33\pi}{4} = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}$.

Exam­ple 5#

Find $\displaystyle \sec\frac{4\pi}{3}$ and $\displaystyle \cot\frac{11\pi}{3}$.

$\displaystyle \frac{4\pi}{3} = 4 \times 60° = 240°$, and

$$\sec 240° = \sec(180° + 60°) = -\sec 60° = -2$$

$\displaystyle \frac{11\pi}{3} = 11 \times 60° = 660° = 360° + 300°$, and

$$\displaystyle \cot 660° = \cot 300° = \cot(360° - 60°) = -\cot 60° = -\frac{1}{\sqrt{3}}$$

Angles on the axes#

For angles that are mul­ti­ples of $90°$, the point $P$ at dis­tance $1$ from $O$ lies on an axis: $(1, 0)$ at $0°$, $(0, 1)$ at $90°$, $(-1, 0)$ at $180°$, $(0, -1)$ at $270°$. Since $\displaystyle \sin\theta = \frac{y}{r}$ and $\displaystyle \cos\theta = \frac{x}{r}$ with $r = 1$, read off the coor­di­nates.

The pat­tern repeats every $360°$.

Where $\sin$, $\cos$ and $\tan$ are zero#

From the table:

• $\sin\theta = 0 \implies \theta = n\pi$, $n \in \mathbb{Z}$ (the mul­ti­ples of $180°$).
• $\displaystyle \cos\theta = 0 \implies \theta = (2n + 1)\frac{\pi}{2}$, $n \in \mathbb{Z}$ (the odd mul­ti­ples of $90°$).
• $\tan\theta = 0 \implies \theta = n\pi$, $n \in \mathbb{Z}$.

So $\sin\pi = \sin 2\pi = \sin 3\pi = 0$, and also $\sin(-\pi) = \sin(-2\pi) = 0$. Like­wise $\displaystyle \cos\frac{\pi}{2} = \cos\frac{3\pi}{2} = \cos\frac{5\pi}{2} = \cos\frac{7\pi}{2} = 0$.

Exam­ple 6#

Eval­u­ate $\sin 120° \cos 150° + \cos 240° \sin 330°$.

$\displaystyle \sin 120° = \frac{\sqrt{3}}{2}$, $\displaystyle \cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}$, $\displaystyle \cos 240° = -\frac{1}{2}$, $\displaystyle \sin 330° = -\sin 30° = -\frac{1}{2}$. So

$$\displaystyle \frac{\sqrt{3}}{2}\left(-\frac{\sqrt{3}}{2}\right) + \left(-\frac{1}{2}\right)\left(-\frac{1}{2}\right) = -\frac{3}{4} + \frac{1}{4} = -\frac{1}{2}$$

Prac­tice#

Find the value of each.

1. $\sin 135°$
2. $\cos 150°$
3. $\tan 240°$
4. $\sec 300°$
5. $\csc 210°$
6. $\cot 315°$
7. $\sin 1110°$
8. $\cos 780°$
9. $\displaystyle \tan\frac{19\pi}{6}$
10. $\sin(-330°)$
11. $\displaystyle \cos\frac{25\pi}{3}$
12. $\sin 810°$ and $\cos 900°$

Answers#