Exact values of trig expressions using reference angles and special angles

Expert intro: These are classic special-angle questions. The main trick is to reduce each angle to a reference angle and keep the sign from the quadrant.

Detailed walkthrough

Use reference angles and quadrant signs.

a) $\cos 330^\circ$

$330^\circ=360^\circ-30^\circ$, so

$\cos 330^\circ=\cos 30^\circ=\frac{\sqrt{3}}{2}.$

b) $\sin(-210^\circ)$

Use odd symmetry:

$\sin(-\theta)=-\sin\theta.$

So

$\sin(-210^\circ)=-\sin 210^\circ.$

Now $210^\circ=180^\circ+30^\circ$, hence

$\sin 210^\circ=-\sin 30^\circ=-\frac12.$

Therefore,

$\sin(-210^\circ)=\frac12.$

c) $\sin 120^\circ$

$120^\circ=180^\circ-60^\circ$, so

$\sin 120^\circ=\sin 60^\circ=\frac{\sqrt{3}}{2}.$

d) $\tan(-300^\circ)$

$\tan$ is odd, so

$\tan(-300^\circ)=-\tan 300^\circ.$

And $300^\circ=360^\circ-60^\circ$, so

$\tan 300^\circ=-\tan 60^\circ=-\sqrt{3}.$

Thus,

$\tan(-300^\circ)=\sqrt{3}.$

e) $\tan 510^\circ$

Subtract $360^\circ$:

$510^\circ-360^\circ=150^\circ,$

so

$\tan 510^\circ=\tan 150^\circ.$

Now $150^\circ=180^\circ-30^\circ$, hence

$\tan 150^\circ=-\tan 30^\circ=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}.$

f) $\cos(-315^\circ)$

$\cos$ is even, so

$\cos(-315^\circ)=\cos 315^\circ.$

And $315^\circ=360^\circ-45^\circ$, therefore

$\cos 315^\circ=\cos 45^\circ=\frac{\sqrt{2}}{2}.$

Answers

• a) $\frac{\sqrt{3}}{2}$
• b) $\frac12$
• c) $\frac{\sqrt{3}}{2}$
• d) $\sqrt{3}$
• e) $-\frac{\sqrt{3}}{3}$
• f) $\frac{\sqrt{2}}{2}$

💡 Pitfall guide

The biggest trap is sign handling. Students often find the reference angle correctly but forget whether the angle is in quadrant II, III, or IV. Also, for negative angles, use symmetry first if that makes the work cleaner; it prevents accidental sign flips.

🔄 Real-world variant

If the same expressions were written in radians, the method would not change: reduce to a standard angle, use even/odd identities, then apply the known special-angle values. The exact values would still come from $\pi/6$, $\pi/4$, and $\pi/3$ type angles.

🔍 Related terms

reference angle, unit circle, quadrant signs