Trigonometry

Understanding the Unit Circle

The unit circle has radius 1 and center at the origin. An angle theta reaches the point (cos(theta), sin(theta)), making the circle a map of trigonometric values.

Reviewed 2026-07-26

Coordinates carry the trigonometric values

At angle theta, the point on the unit circle is (cos(theta), sin(theta)). Signs follow the quadrant automatically.

The four axis points anchor the signs: (1,0), (0,1), (-1,0), and (0,-1). Reference angles supply the coordinate magnitudes between them.

Coordinates become trigonometric values

Start on the positive x-axis and rotate counterclockwise for positive angles. The x-coordinate of the endpoint is cosine and the y-coordinate is sine.

Tangent is y/x where x is nonzero. This coordinate view explains both magnitude and sign across all four quadrants.

FormulaP(theta) = (cos(theta), sin(theta)); tan(theta)=y/x

First-quadrant coordinates worth deriving once

The remaining quadrants use the same magnitudes with signs determined by x and y.

Angle	Radians	cos(theta)	sin(theta)
0 degrees	0	1	0
30 degrees	pi/6	sqrt(3)/2	1/2
45 degrees	pi/4	sqrt(2)/2	sqrt(2)/2
60 degrees	pi/3	1/2	sqrt(3)/2
90 degrees	pi/2	0	1

Worked example 1

Find values at 150 degrees

The reference angle is 30 degrees in quadrant II.

cos magnitude = sqrt(3)/2 and is negative
sin magnitude = 1/2 and is positive
tan = sin/cos

Result: cos150=-sqrt(3)/2, sin150=1/2, tan150=-sqrt(3)/3

Quadrants and signs

In quadrant I, both coordinates are positive. Quadrant II has negative cosine and positive sine; quadrant III has both negative; quadrant IV has positive cosine and negative sine.

Reference angles let you reuse the same coordinate magnitudes while assigning signs according to the quadrant.

QI: sin +, cos +
QII: sin +, cos -
QIII: sin -, cos -
QIV: sin -, cos +

Use a reference angle and a quadrant

Separate the magnitude from the signs instead of memorizing every coordinate.

1. Find cos(210 degrees) and sin(210 degrees).

Hint: The reference angle is 30 degrees in quadrant III.

Answer: cos = -sqrt(3)/2 and sin = -1/2

2. Which angle has point (0, -1)?

Hint: Locate the negative y-axis.

Answer: 270 degrees or 3pi/2 radians

Questions about the unit circle

Why must the radius be 1?

A unit radius makes the coordinate ratios equal directly to sine and cosine.