The unit circle & exact values
On the unit circle, a point at angle θ has coordinates (cos θ, sin θ), which extends the ratios to angles beyond 90° and to negative angles. The signs follow the quadrants — remembered as ASTC, "All Students Take Calculus": all positive in Q1, only sine in Q2, only tangent in Q3, only cosine in Q4.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Locate the quadrantReduce the angle to [0°, 360°) by adding or subtracting full turns, then see which quadrant it lands in.
- Find the reference angleThe acute angle to the nearest part of the x-axis. Its exact value gives the magnitude.
- Attach the sign from ASTCThe reference angle supplies the number; the quadrant supplies the sign.
- Know the quadrantal valuescos 0° = 1, sin 90° = 1, cos 180° = −1, sin 270° = −1, and tan is undefined at 90° and 270°.
Worked example
Find the exact value of cos 240° and sin 240°.
- 240° is between 180° and 270°, so it lies in the third quadrant.
- Reference angle: 240° − 180° = 60°.
- In Q3 both sine and cosine are negative (only tangent is positive).
- cos 60° = 1/2 and sin 60° = √3/2.
Answer. cos 240° = −1/2 and sin 240° = −√3/2.
Where marks get dropped
These are the specific errors that cost credit on the unit circle & exact values questions — QED's rubric penalises each of them separately.
- Measuring the reference angle to the y-axis. It is always to the nearest part of the x-axis.
- Getting the quadrant signs wrong — in Q2 only sine is positive, and in Q3 only tangent.
- Giving a decimal where an exact surd is required. −√3/2, not −0.866.
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The unit circle & exact values — frequently asked questions
Why is it called the unit circle?
Because it has radius 1, which makes the coordinates equal to cosine and sine directly — with any other radius you would divide by r.
How do negative angles work?
They are measured clockwise. cos(−θ) = cos θ since cosine is even, and sin(−θ) = −sin θ since sine is odd.
Where does sin²θ + cos²θ = 1 come from?
Directly from Pythagoras applied to the point (cos θ, sin θ) on a circle of radius 1.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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