Unit Circle & Angle Measure

Right-triangle trigonometry is limited to angles between 0° and 90°. The unit circle extends sine and cosine to all angles, opening the door to modeling waves, rotations, and periodic phenomena.

The Unit Circle

The unit circle is a circle of radius 1 centered at the origin. An angle θ is measured from the positive x-axis, counter-clockwise. The point where the terminal side of the angle meets the circle has coordinates:

(cos θ, sin θ)

This means: cos θ is the x-coordinate and sin θ is the y-coordinate of the point on the unit circle.

Radians vs. Degrees

A radian measures angle by arc length: an angle of 1 radian subtends an arc of length 1 on the unit circle. Since the full circumference is 2π, a full revolution is 2π radians.

360° = 2π rad 180° = π rad

Conversion Formulas

Degrees to Radians:

radians = degrees \times π/180

Radians to Degrees:

degrees = radians \times 180/π

Worked Example 1 -- Converting

Convert 150° to radians and π/3 to degrees.

150° × π/180 = 150π/180 = 5π/6 radians.
π/3 × 180/π = 180/3 = 60°.

Key Angles on the Unit Circle

Degrees	Radians	cos θ	sin θ
0°	0	1	0
30°	π/6	√3/2	1/2
45°	π/4	√2/2	√2/2
60°	π/3	1/2	√3/2
90°	π/2	0	1
180°	π	-1	0
270°	3π/2	0	-1
360°	2π	1	0

Memorizing the First Quadrant

For angles 0°, 30°, 45°, 60°, 90°, the sine values follow the pattern √0/2, √1/2, √2/2, √3/2, √4/2 -- that is, 0, 1/2, √2/2, √3/2, 1. The cosine values are the same sequence in reverse order.

Worked Example 2 -- Coordinates on the Unit Circle

Find the exact coordinates of the point at 225° on the unit circle.

225° = 180° + 45°, so the reference angle is 45°. The point is in Quadrant III.
In Q III, both x and y are negative: (cos 225°, sin 225°) = (-√2/2, -√2/2).

Worked Example 3 -- Evaluating with the Unit Circle

Find sin(5π/6) and cos(5π/6).

5π/6 radians = 150°. Reference angle = 180° - 150° = 30°. Quadrant II.
sin(150°) = sin(30°) = 1/2 (positive in Q II).
cos(150°) = -cos(30°) = -√3/2 (negative in Q II).

Common Mistake

Forgetting to check the quadrant when using reference angles. The reference angle gives the magnitude of sin and cos, but the sign depends on the quadrant: All positive in Q I, Sine in Q II, Tangent in Q III, Cosine in Q IV (mnemonic: All Students Take Calculus).

Practice Problems

Convert 240° to radians.
Show Solution

240 × π/180 = 4π/3 radians.
Convert 7π/4 to degrees.
Show Solution

7π/4 × 180/π = 7 × 45 = 315°.
Find sin(120°) and cos(120°) exactly.
Show Solution

Reference angle = 60°, Q II. sin(120°) = √3/2, cos(120°) = -1/2.
What are the coordinates of the point at 3π/4 on the unit circle?
Show Solution

3π/4 = 135°. Reference angle = 45°, Q II. Coordinates: (-√2/2, √2/2).
An angle of 2 radians -- is it larger or smaller than 90°? Explain.
Show Solution

2 rad × 180/π ≈ 114.6°. It is larger than 90° -- the angle lies in Quadrant II.

Summary

The unit circle defines cos θ and sin θ as coordinates of a point.
Radians measure angle by arc length; 180° = π radians.
Key angles and their coordinates should be memorized.
Reference angles + quadrant signs let you evaluate trig functions at any angle.