Math · Precalculus · Concept

The unit circle, radians and exact trig values

The unit circle has radius 1 and its center at the origin. An angle θ in standard position ends at the point (cos θ, sin θ), so sine and cosine are coordinates and tangent is their ratio. Radians measure angles by arc length, and a reference angle with the signs of its quadrant gives the exact values at the special angles.

Radians measure arc length

On a circle of radius r, an angle of θ radians cuts off an arc of length s = rθ. A full turn is 2π radians, so 180° = π radians. To convert, multiply degrees by π/180, or radians by 180/π. Calculus works in radians: the derivative of sin x is cos x only when x is in radians.

s = r θ, 180^{\circ} = π rad

Sine and cosine are coordinates

Start at (1, 0) and turn counterclockwise through θ, or clockwise if θ is negative. The turn ends at the point (cos θ, sin θ). Tangent is the ratio sin θ/cos θ, the slope of the line from the origin to that point, and it is undefined where cos θ = 0. Because the point lies on the circle x² + y² = 1, cos²θ + sin²θ = 1 for every θ.

(x, y) = (cos θ, sin θ), tan θ = \frac{sin θ}{cos θ}

\begin{array}{c} (x, y) = (cos θ, sin θ) \\ tan θ = \frac{sin θ}{cos θ} \end{array}

The special angles

Three first-quadrant angles have exact coordinates that come from two triangles: half a square gives 45°, and half an equilateral triangle gives 30° and 60°. Every other special angle reuses these values with new signs.

Exact values in the first quadrant
Angle	cos θ	sin θ	tan θ
0	1	0	0
π/6 (30°)	√3/2	1/2	√3/3
π/4 (45°)	√2/2	√2/2	1
π/3 (60°)	1/2	√3/2	√3
π/2 (90°)	0	1	undefined

Reference angles and signs

The reference angle is the acute angle between the terminal side and the x-axis. It fixes the size of each value, and the quadrant fixes the sign: cosine takes the sign of x and sine the sign of y. All three functions are positive in quadrant I, only sine in quadrant II, only tangent in quadrant III and only cosine in quadrant IV.

Reference angle and positive functions for 0 < θ < 2π
Quadrant	Reference angle	Positive
I	θ	sin, cos, tan
II	π − θ	sin
III	θ − π	tan
IV	2π − θ	cos

Coterminal angles and periods

Angles that differ by a whole number of turns end at the same point, so sine and cosine repeat every 2π: they are periodic with period 2π. Tangent repeats every π, because the point opposite (x, y) on the circle is (−x, −y), which has the same ratio.

sin (θ + 2 π) = sin θ, tan (θ + π) = tan θ

\begin{array}{c} sin (θ + 2 π) = sin θ \\ tan (θ + π) = tan θ \end{array}

Negative angles and symmetry

A negative angle turns clockwise, which reflects the point across the x-axis: the x-coordinate stays and the y-coordinate changes sign. So cosine is an even function and sine is an odd one.

cos (- θ) = cos θ, sin (- θ) = - sin θ

\begin{array}{c} cos (- θ) = cos θ \\ sin (- θ) = - sin θ \end{array}

From the circle to the graphs

Plotting sin x against x unrolls the circle into a wave that repeats every 2π and stays between −1 and 1. Transformations reshape it: in y = a sin(bx), |a| is the amplitude and 2π/|b| is the period.

y = a sin (b x) : period \frac{2 π}{∣ b ∣}

Common mistakes

Evaluating a radian angle in degree mode: sin(π/6) is 1/2, but degree mode returns about 0.0091.
Taking the sign from the reference angle, which is always acute: the quadrant sets the sign.
Swapping the coordinates: cosine is the x-coordinate and sine the y-coordinate.
Giving tan(π/2) a value: cos(π/2) = 0, so tangent is undefined there.

Key terms

Unit circle: The circle of radius 1 centered at the origin. An angle θ in standard position meets it at the point (cos θ, sin θ), which defines sine and cosine for every real angle.
Radian: An angle measure: arc length divided by radius. A full turn is 2π radians (360°), and calculus formulas such as (sin x)′ = cos x assume radians.
Reference angle: The acute angle between the terminal side of an angle and the x-axis. The trig values of an angle equal those of its reference angle, up to a sign set by the quadrant.
Coterminal angles: Angles in standard position that share a terminal side. They differ by a whole number of full turns, 2πk radians or 360k°, so they have the same trig values.
Sine function: On the unit circle, sin θ is the y-coordinate of the point at angle θ. Its values always lie between −1 and 1, and they repeat every 2π radians (360°).
Cosine function: On the unit circle, cos θ is the x-coordinate of the point at angle θ. It lies between −1 and 1, and sin²θ + cos²θ = 1 for every angle.
Tangent function: tan θ = sin θ/cos θ, defined wherever cos θ ≠ 0. On the unit circle it is the slope of the line through the origin at angle θ; it is not the same thing as a tangent line to a graph.
Periodic function: A function that repeats after a fixed shift T: f(x + T) = f(x) for every x. The smallest such T is the period, such as 2π for sin x.

Work through an example

Find the exact values of sin θ, cos θ and tan θ for θ = 2π/3.

Find exact trig values with a reference angle →

Convert between degrees and radians →

Evaluate trig at negative and large angles →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Make it concrete

Try in the workspace

Open the example inputs, change a value and keep a useful result on your board.

Open the Unit Circle in Math Reference Check the values in Math See sine and cosine as graphs Open worked example on a board Radians and arc length in Math Reference

Your existing work stays on this device. Examples open as editable copies.