Curriculum fit: US high school Precalculus and Algebra 2 (Common Core Functions: Trigonometric Functions) and UK A-Level Maths (radian measure). GCSE Maths needs exact trig values for 0° to 90° but not radians.
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What Is It?
The unit circle is a circle with radius 1 centred at the origin of a graph. Start at the positive x-axis and turn anticlockwise by an angle θ. The point where you stop on the circle has coordinates (cos θ, sin θ). So the x-coordinate is the cosine and the y-coordinate is the sine.
Angles can be measured in degrees or in radians. One radian is the angle at the centre of a circle that cuts off an arc equal in length to the radius. A full turn is 360°, which is 2π radians, so half a turn is 180°, which is π radians. Radians make formulas for arcs, sectors and calculus simpler, which is why A-Level and college courses use them.
Key Formulas and Ideas
- Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π.
- Point on the unit circle at angle θ: (cos θ, sin θ), so sin²θ + cos²θ = 1.
- tan θ = sin θ ÷ cos θ, which is undefined when cos θ = 0 (at 90° and 270°).
- Exact values: 30° = π/6 gives (√3/2, 1/2); 45° = π/4 gives (√2/2, √2/2); 60° = π/3 gives (1/2, √3/2); 90° = π/2 gives (0, 1).
- Signs by quadrant: Quadrant I all positive, II only sin, III only tan, IV only cos (remember CAST or ASTC).
- Arc length s = rθ and sector area A = ½r²θ, with θ in radians.
Worked Examples
Converting Between Degrees and Radians
Convert 150° to radians, and 3π/4 radians to degrees.
- For degrees to radians multiply by π/180: 150 × π/180 = 150π/180.
- Simplify by dividing top and bottom by 30: 5π/6.
- For radians to degrees multiply by 180/π: (3π/4) × (180/π) = 3 × 45.
- 3 × 45 = 135.
✓ Answer 150° = 5π/6 radians and 3π/4 radians = 135°
Exact Value in Another Quadrant
Find the exact values of sin 210° and cos 210°.
- 210° is in Quadrant III, 30° past 180°, so the reference angle is 30°.
- sin 30° = 1/2 and cos 30° = √3/2.
- In Quadrant III both sine and cosine are negative.
- So sin 210° = −1/2 and cos 210° = −√3/2.
✓ Answer sin 210° = −1/2 and cos 210° = −√3/2
Arc Length and Sector Area
A sector has radius 6 cm and angle 2π/3 radians. Find the arc length and the sector area.
- Arc length s = rθ = 6 × 2π/3 = 4π.
- 4π ≈ 12.57 cm.
- Sector area A = ½r²θ = ½ × 36 × 2π/3 = 12π.
- 12π ≈ 37.70 cm².
✓ Answer Arc length 4π ≈ 12.57 cm and area 12π ≈ 37.70 cm²
Common Mistakes to Avoid
- Leaving the calculator in degree mode when the question is in radians, or the other way round.
- Swapping sine and cosine. On the unit circle, cosine is x and sine is y.
- Forgetting the sign in quadrants II, III and IV. Find the reference angle first, then apply the sign.
- Saying tan 90° = 1 or 0. It is undefined because you would divide by cos 90° = 0.
- Using the arc-length and sector formulas with degrees. They only work with radians.
How to Remember the Exact Values
You only need one pattern. For 0°, 30°, 45°, 60° and 90° the sine values are √0/2, √1/2, √2/2, √3/2 and √4/2. That is 0, 1/2, √2/2, √3/2 and 1.
The cosine values are the same list in reverse order: 1, √3/2, √2/2, 1/2 and 0. Then use the reference angle and the quadrant signs to get the values for every other multiple of 30° and 45°.
Practice Questions
Frequently Asked Questions
Why do we use radians instead of degrees?
Radians connect an angle directly to a length, so s = rθ and A = ½r²θ have no extra constants. In calculus, the derivative of sin x is cos x only when x is in radians. Degrees are fine for everyday angles.
Is the unit circle on the GCSE?
GCSE Maths in England asks you to know the exact values of sin and cos for 0°, 30°, 45°, 60° and 90°, and of tan for 0°, 30°, 45° and 60°. Radians and the full unit circle come later, in A-Level Maths.
What is the unit circle used for?
It defines sine, cosine and tangent for any angle, not just angles inside a right-angled triangle. This lets you solve trigonometric equations and graph the sine and cosine waves.
Keep Going With These Guides
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Solve non-right-angled triangles with the sine and cosine rules (law of sines and law of cosines) and find a triangle's area, with examples.
GCSEA-LevelHigh SchoolCircle Theorems Explained: Rules, Examples and Reasons
Learn the main circle theorems and use them to find angles with a reason for every step. GCSE Higher and high school Geometry, with worked examples.
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Learn the area and circumference formulas for circles, how to use radius and diameter, and how to work with semicircles, with worked examples.
KS3GCSEMiddle SchoolHigh SchoolSources and Review
- Common Core State Standards: Mathematics (high school)
- DfE: GCSE mathematics subject content and assessment objectives
Reviewed by StudyMorrow editorial review on 7 Nov 2026.