Unit Circle

How to read the unit circle

The unit circle has radius 1, centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the point where the angle's ray meets the circle is (cos θ, sin θ) — the x-coordinate is the cosine, the y-coordinate is the sine. Tap the circle above and watch the dashed lines: the horizontal drop to the x-axis is cos θ, the vertical drop to the y-axis is sin θ.

1. Tap the circle to pick any angle, or type degrees or radians — even fractions of π like 3π/4.
2. Read the coordinates. The point shown is (cos θ, sin θ): that single fact answers most unit-circle quiz questions.
3. Check the exact value. For the 16 standard angles the readout shows the memorization form (√2/2, −√3/3…) alongside the decimal.
4. Use the reference angle for non-standard angles: find the trig value of the acute reference angle, then apply the quadrant sign.

The quadrant sign rules (ASTC)

Because sin is a y-coordinate and cos is an x-coordinate, their signs follow the quadrant: All positive in I, only Sine positive in II, only Tangent positive in III, only Cosine positive in IV — "All Students Take Calculus". The readout above names the quadrant for every tap.

What is the unit circle?

The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured from the positive x-axis, the point on the circle has coordinates (cos θ, sin θ). It turns trigonometry into geometry: every trig value is just a coordinate.

How do sin and cos relate to the unit circle?

On the unit circle, cos θ is the x-coordinate of the point at angle θ and sin θ is the y-coordinate. The dashed lines on the circle above show exactly this: the horizontal drop is cos θ, the vertical drop is sin θ.

Why is tan undefined at 90° and 270°?

Because tan θ = sin θ / cos θ, and at 90° and 270° the point sits on the y-axis where cos θ = 0. Dividing by zero is undefined, so tan has vertical asymptotes there. The table marks both as undefined.

How do I find the reference angle?

The reference angle is the acute angle between the terminal side and the x-axis: in quadrant I it is θ itself, in II it is 180° − θ, in III it is θ − 180°, and in IV it is 360° − θ. Angles on an axis have no reference angle. The readout above computes it for you.

How do I convert degrees to radians?

Multiply degrees by π/180. So 90° = π/2, 180° = π, and 360° = 2π. Going the other way, multiply radians by 180/π. Both inputs above accept either unit, and you can even type fractions of π like 3π/4.

How do I remember the signs in each quadrant?

Use the mnemonic All Students Take Calculus: in quadrant I All functions are positive, in II only Sine is positive, in III only Tangent is positive, and in IV only Cosine is positive. It follows directly from the signs of the x and y coordinates on the circle.