Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 radians and unit circle? Then this Algebra 2 lesson on Radians and the Unit Circle Explained explains each idea step by step, with worked examples.
Quick answer: A radian is the angle that cuts off an arc equal to the radius, so 180° equals π radians. On the unit circle, the point at angle θ is (cos θ, sin θ). The identity sin² θ + cos² θ = 1 gives one ratio from the other once the quadrant is known.
Standards: F-TF.A.1, F-TF.A.2, F-TF.C.8 · Level: Grade 11 Algebra 2
What is a radian?
A radian measures an angle by the arc it cuts off on a circle, compared with the radius. Because the full circle has arc 2πr, a full turn is 2π radians.
π radians = 180°. Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Arc length: s = rθ with θ in radians.
How does the unit circle define sine and cosine?
The unit circle has radius 1 and center (0, 0). Rotate counterclockwise by θ from the positive x-axis. The endpoint is (cos θ, sin θ), and tan θ = sin θ ÷ cos θ.
| θ | 0 | π/6 | π/4 | π/3 | π/2 |
|---|---|---|---|---|---|
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
Other angles use a reference angle, the acute angle to the x-axis, and the quadrant sign.
How do you find one ratio from another?
sin2 θ + cos2 θ = 1
- Substitute the known ratio into the identity.
- Solve for the other ratio squared, then take the square root.
- Choose the sign from the quadrant.
If sin θ = 7/25 and θ is in quadrant II, then cos2 θ = 1 − 49/625 = 576/625. So cos θ = −24/25, since cosine is negative in quadrant II, and tan θ = −7/24.
Forgetting the sign after taking the square root. The identity gives only the size of the ratio.
For 30° and 60°, remember that the smaller angle has the smaller sine: sin 30° = 1/2 and sin 60° = √3/2.
- π radians = 180°, so a full turn is 2π radians.
- Arc length is s = rθ when θ is in radians.
- The terminal point of angle θ on the unit circle is (cos θ, sin θ).
- sin² θ + cos² θ = 1 for every angle θ.
- Reference angles and quadrant signs give the exact value of any special angle.
Radians and the Unit Circle Explained: frequently asked questions
Why do mathematicians use radians instead of degrees?
Radians connect angle to arc length directly with s = rθ, and they make formulas in calculus and trigonometry simpler. A degree is an arbitrary one 360th of a turn, while a radian comes from the circle itself.
How do I convert degrees to radians?
Multiply the number of degrees by π/180. For example, 150° × π/180 = 5π/6. To go the other way multiply radians by 180/π.
What is a reference angle?
It is the acute angle between the terminal side and the x-axis. Values of sine and cosine for any angle equal those of the reference angle, up to a sign set by the quadrant.
How can I tell the sign of sine or cosine?
Sine is the y-coordinate, so it is positive above the x-axis (quadrants I and II). Cosine is the x-coordinate, so it is positive to the right of the y-axis (quadrants I and IV).
Why does sin² θ + cos² θ = 1?
The point (cos θ, sin θ) lies on the unit circle, whose equation is x² + y² = 1. Substituting the coordinates gives the identity, which is the Pythagorean Theorem in disguise.