Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for precalculus unit circle? Then this Precalculus lesson on Radians, Arc Length and the Unit Circle explains each idea step by step, with worked examples.
Quick answer: A radian measures an angle by the arc it cuts on a circle of radius 1, so 180 degrees equals pi radians. On the unit circle, the point at angle theta is (cos theta, sin theta), and reference angles with symmetry give exact sine, cosine and tangent values at multiples of pi/6, pi/4 and pi/3.
Standards: F-TF.A.1, F-TF.A.2, F-TF.A.3, F-TF.A.4 · Level: Grade 12 Precalculus
What is a radian, and how do you convert to degrees?
A radian is the angle at the center of a circle that cuts off an arc exactly as long as the radius. A full turn is 2π radians, so 180° = π radians. Multiply degrees by π/180 to get radians, and radians by 180/π to get degrees.
- Arc length: s = rθ (θ in radians).
- Angular speed: ω = θ/t, in radians per unit of time.
- Linear speed of a point on the rim: v = rω.
How do you read the unit circle?
On the unit circle (radius 1, center at the origin), the point reached after turning through θ from the positive x-axis is (cos θ, sin θ). Then tan θ = sin θ ÷ cos θ. Memorize the first-quadrant values and obtain every other special angle by symmetry.
| Angle | cos | sin | tan |
|---|---|---|---|
| π/6 | √3/2 | 1/2 | √3/3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | 1/2 | √3/2 | √3 |
How do reference angles and symmetry give exact values?
- Subtract multiples of 2π until the angle lies in [0, 2π).
- Find the quadrant and the reference angle α, the acute angle to the x-axis.
- Use the value at α and attach the sign for the quadrant: cosine is positive in Quadrants I and IV, sine in I and II.
Find cos(7π/6) and sin(7π/6). The angle is in Quadrant III, and 7π/6 − π = π/6 is the reference angle. Both coordinates are negative, so cos(7π/6) = −√3/2 and sin(7π/6) = −1/2.
Sine and cosine repeat every 2π, and tangent every π. Sine is odd, sin(−θ) = −sin θ, while cosine is even, cos(−θ) = cos θ.
Forgetting the sign. The reference angle only gives the size of the value; the quadrant decides whether it is positive or negative.
Picture the point on the circle first. If it is left of the y-axis, cosine is negative; if it is below the x-axis, sine is negative.
- Degrees to radians: multiply by π/180; radians to degrees: multiply by 180/π.
- Arc length s = rθ and linear speed v = rω, with θ in radians.
- The point at angle θ on the unit circle is (cos θ, sin θ).
- Sine is odd, cosine is even; sine and cosine have period 2π, tangent has period π.
- Reference angle gives the size of a value; the quadrant gives its sign.
Radians, Arc Length and the Unit Circle: frequently asked questions
What is a radian?
It is the angle at the center of a circle that intercepts an arc equal in length to the radius. One radian is about 57.3 degrees, and a full circle is 2π radians.
How do you convert degrees to radians?
Multiply by π/180. For example, 150° × π/180 = 5π/6. To go back, multiply radians by 180/π, so 5π/6 × 180/π = 150°.
How do you find exact trig values at special angles?
Reduce the angle to [0, 2π), find its reference angle, recall the value at π/6, π/4 or π/3, and then choose the sign from the quadrant where the terminal side lies.
What is a reference angle?
It is the acute angle between the terminal side of an angle and the nearest part of the x-axis. For 4π/3, the terminal side is π/3 past the negative x-axis, so the reference angle is π/3.
Why is cosine even and sine odd?
Turning by −θ reflects the point across the x-axis. The x-coordinate stays the same, so cos(−θ) = cos θ, while the y-coordinate changes sign, so sin(−θ) = −sin θ.
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