Unit circle with exact values
Drag a point round the unit circle to read sin, cos and tan as exact values at every special angle, in degrees and radians. For GCSE Higher and A-level maths trigonometry.
0°
0 rad
On the positive x-axis
The point is at (cos θ, sin θ): (1, 0)
Sine and cosine graphs
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Learn this properly
Lessons from the Maths course on this topic: worked steps, then exam-style questions with new numbers every time.
The idea
Measure the angle θ anticlockwise from the positive x-axis. The radius joins the centre to the point. It is the hypotenuse of a right-angled triangle, and its length is 1.
The triangle's horizontal side is 1 × cos θ and its vertical side is 1 × sin θ. So the point on the circle is (cos θ, sin θ), and tan θ = sin θ ÷ cos θ.
Exact values from 0° to 90°
Learn these five rows. Every other special angle on the circle uses the same numbers with a different sign.
| Angle | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | 3/2 | 3/3 |
| 45° | π/4 | 2/2 | 2/2 | 1 |
| 60° | π/3 | 3/2 | 1/2 | 3 |
| 90° | π/2 | 1 | 0 | undefined |
You will also see tan 30° written as 1/3 and sin 45° as 1/2. They are the same numbers: 1/3 = 3/3 and 1/2 = 2/2.
Every special angle round the circle
Past 90°, find the reference angle (the acute angle between the radius and the x-axis), read its values from the table above, then give each one the sign for its quadrant.
| Angle | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 120° | 2π/3 | 3/2 | −1/2 | −3 |
| 135° | 3π/4 | 2/2 | −2/2 | −1 |
| 150° | 5π/6 | 1/2 | −3/2 | −3/3 |
| 180° | π | 0 | −1 | 0 |
| 210° | 7π/6 | −1/2 | −3/2 | 3/3 |
| 225° | 5π/4 | −2/2 | −2/2 | 1 |
| 240° | 4π/3 | −3/2 | −1/2 | 3 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 300° | 5π/3 | −3/2 | 1/2 | −3 |
| 315° | 7π/4 | −2/2 | 2/2 | −1 |
| 330° | 11π/6 | −1/2 | 3/2 | −3/3 |
| 360° | 2π | 0 | 1 | 0 |
Key facts
- Point on the circle: (cos θ, sin θ). x is cos θ and y is sin θ.
- Tangent: tan θ = sin θ ÷ cos θ. It is undefined at 90° and 270°, because cos θ is 0 there and you cannot divide by 0.
- Identity: sin2θ + cos2θ = 1 for every angle. It is Pythagoras on the triangle, whose hypotenuse is 1.
- Signs (CAST): all three ratios are positive in the first quadrant, only sin in the second, only tan in the third and only cos in the fourth.
- Symmetry: sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ. Going the other way round, sin(−θ) = −sin θ and cos(−θ) = cos θ.
- Period: the values repeat every 360°, so sin(θ + 360°) = sin θ and cos(θ + 360°) = cos θ.
- Degrees to radians: multiply by π/180, so 30° = π/6 and 135° = 3π/4. To go back, multiply by 180/π.
Worked examples
Example 1
Find the exact values of sin, cos and tan at 150°.
- 150° is in the second quadrant, where sin is positive and cos is negative.
- The reference angle is 180° − 150° = 30°.
- sin 30° = 1/2 and cos 30° = 3/2.
- So sin 150° = 1/2, cos 150° = −3/2 and tan 150° = (1/2) ÷ (−3/2) = −1/3 = −3/3.
Answer: sin 150° = 1/2, cos 150° = −3/2, tan 150° = −3/3.
Example 2
Solve sin θ = 1/2 for 0° ≤ θ < 360°.
- sin θ is positive, so θ is in the first or the second quadrant.
- sin 30° = 1/2, so the reference angle is 30°.
- First quadrant: θ = 30°. Second quadrant: θ = 180° − 30° = 150°.
Answer: θ = 30° or θ = 150° (π/6 or 5π/6 in radians).
Common mistakes
- Giving cos a positive sign in the second quadrant: only sin is positive there.
- Swapping sin and cos: x is cos θ and y is sin θ.
- Writing a value for tan 90°: it is undefined, because cos 90° = 0.
- Stopping at one answer: sin θ = 1/2 has two solutions between 0° and 360°, 30° and 150°.
- Mixing up the units: check whether the question uses degrees or radians, and set your calculator to match. 150° is 5π/6 radians.
FAQ
- Why is the point at (cos θ, sin θ)?
- The radius is the hypotenuse of the right-angled triangle and it has length 1. The horizontal side is 1 × cos θ and the vertical side is 1 × sin θ, so those are the point's coordinates.
- Where do exact values like √3/2 come from?
- From two special triangles. Half of an equilateral triangle with sides 2 gives sides 1, 3 and 2, so sin 60° = 3/2 and cos 60° = 1/2. A right-angled isosceles triangle with legs 1 has hypotenuse 2, so sin 45° = cos 45° = 1/2 = 2/2.
- What is a reference angle?
- It is the acute angle between the radius and the x-axis. An angle and its reference angle have the same sin, cos and tan apart from the sign, which depends on the quadrant.
- Why is tan 90° undefined?
- tan θ = sin θ ÷ cos θ, and at 90° cos θ = 0, so tan 90° would mean dividing 1 by 0, which has no value. The graph of tan θ has a vertical asymptote at 90°, and the same happens at 270°.
- Which exact values do I need for GCSE?
- The content that AQA, Pearson Edexcel and OCR all follow (DfE G21) asks you to know sin and cos of 0°, 30°, 45°, 60° and 90°, and tan of 0°, 30°, 45° and 60°, without a calculator. That is the first table on this page, and it is on both Foundation and Higher papers.
- Why does sin 30° = cos 60°?
- In a right-angled triangle the other two angles add up to 90°, so the side opposite 30° is the side adjacent to 60°. That makes sin 30° and cos 60° the same ratio, 1/2. In general sin θ = cos(90° − θ), which is why sin 60° = cos 30° = 3/2 as well.