Overview

Most contest trig starts with the unit circle. You should be able to convert between degrees and radians instantly and recall exact values for special angles.

Core Conversions

• $180^\circ = \pi$ radians.
• $360^\circ = 2\pi$ radians.
• $1^\circ = \pi/180$ radians.

Common angles:

• $30^\circ = \pi/6$, $45^\circ = \pi/4$, $60^\circ = \pi/3$.
• $90^\circ = \pi/2$, $120^\circ = 2\pi/3$, $135^\circ = 3\pi/4$.
• $150^\circ = 5\pi/6$, $180^\circ = \pi$.

Unit Circle Definition

For angle $\theta$, the point on the unit circle is $(\cos\theta, \sin\theta)$. So:

• $\cos\theta$ is the $x$-coordinate.
• $\sin\theta$ is the $y$-coordinate.
• $\tan\theta = \sin\theta/\cos\theta$ when $\cos\theta \ne 0$.

Core Skills

Convert Degrees and Radians

Multiply by $\pi/180$ to convert degrees to radians, and by $180/\pi$ to convert radians to degrees.

Use Quadrants for Signs

Remember the sign of $\sin$ and $\cos$ in each quadrant to avoid sign errors.

Build Other Angles

Use reference angles to find values at $150^\circ$, $210^\circ$, $330^\circ$, and other nonstandard angles.

Must-Know Values

A quick memory trick for $\sin$ at $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$:

$\sin\theta = \frac{\sqrt{0}}{2},\;\frac{\sqrt{1}}{2},\;\frac{\sqrt{2}}{2},\;\frac{\sqrt{3}}{2},\;\frac{\sqrt{4}}{2}.$

For $\cos$, reverse the order.

Reciprocal Functions

• $\csc\theta = 1/\sin\theta$.
• $\sec\theta = 1/\cos\theta$.
• $\cot\theta = 1/\tan\theta = \cos\theta/\sin\theta$.

Worked Example

Evaluate $\sin 30^\circ + \cos 60^\circ + \tan 45^\circ$.

$\sin 30^\circ=1/2$, $\cos 60^\circ=1/2$, $\tan 45^\circ=1$. Sum is $2$.

More Examples

Example 1: Quadrant Sign

Find $\cos 210^\circ$.

Reference angle is $30^\circ$, and cosine is negative in quadrant III, so $\cos 210^\circ = -\sqrt{3}/2$.

Example 2: Convert to Radians

Convert $135^\circ$ to radians.

$135^\circ = 3\pi/4$.

Example 3: Reference Angle

Find $\sin 300^\circ$.

Reference angle is $60^\circ$, and sine is negative in quadrant IV, so $\sin 300^\circ = -\sqrt{3}/2$.

Strategy Checklist

• Convert degrees and radians first if needed.
• Use a reference angle and quadrant signs.
• Recall the special-angle values without a calculator.

Common Pitfalls

• Mixing degrees and radians in the same problem.
• Forgetting the sign of $\sin$ or $\cos$ by quadrant.
• Using $\tan\theta$ when $\cos\theta=0$.

Practice Problems