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 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

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 +

Special-angle coordinates

The 30-60-90 and 45-45-90 triangles supply exact coordinates for pi/6, pi/4, and pi/3. Symmetry then gives corresponding points around the circle.

Memorizing one first-quadrant sequence and understanding symmetry is more reliable than memorizing every point independently.

Common mistakes

Reversing the coordinate order and treating sine as x.
Using first-quadrant signs for every angle.
Forgetting that the same point is reached after adding any full revolution.

Where this idea is used

Finding exact trigonometric values without a decimal calculator.
Understanding periodicity, signs, and inverse-trig ranges.
Representing circular motion and complex-number arguments.

Worked examples

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

Find the angle for (-sqrt2/2, -sqrt2/2)

Equal coordinate magnitudes give a 45-degree reference angle.

Both coordinates are negative, so the point is in quadrant III
180 degrees + 45 degrees = 225 degrees

Result: theta = 225 degrees = 5pi/4 radians, plus full turns

Questions about the unit circle

Why must the radius be 1?

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

How often do points repeat?

After every full turn: 2pi radians or 360 degrees.

Where is tangent undefined?

At points with x=0, because tangent is y/x and division by zero is undefined.