The unit circle is a circle with radius 1 and centre (0, 0). The point on it at angle θ is (cos θ, sin θ). So x is cos θ and y is sin θ.
x first, then y: cos first, then sin.
Hello, curious mind!
What is the unit circle?
The unit circle is the circle x² + y² = 1. Measure an angle θ from the positive x-axis, turning anticlockwise. Where that angle meets the circle, the point is (cos θ, sin θ).
Why is the radius 1?
Draw a right-angled triangle from the centre to the point. Its long side is the radius, 1. So cos θ = adjacent ÷ 1 = x, and sin θ = opposite ÷ 1 = y.
As x² + y² = 1, cos²θ + sin²θ = 1 at every angle. That is the Pythagoras theorem with a long side of 1. And tan θ = y ÷ x, which has no value at 90° and 270°, where x = 0.
Remember this
One full turn is 360°, or 2π radians. So 180° = π, 60° = π/3 and 30° = π/6.
Build the unit circle chart from two triangles
Do not learn 16 points. Two triangles give every value from 0° to 90°. The other quadrants reuse them with a sign.
The 45° triangle
Take a right-angled triangle with two 45° angles and a long side of 1. The short sides are equal, and each is √2/2, because 1/2 + 1/2 = 1. So the 45° point is (√2/2, √2/2).
The 30° and 60° triangle
Cut an equilateral triangle with side 2 in half. You get sides 1, √3 and 2, with angles 30°, 60° and 90°. Halve each side: 1/2, √3/2 and 1. The side 1/2 is opposite 30°. So the 30° point is (√3/2, 1/2) and the 60° point is (1/2, √3/2).
| Angle θ | Point (cos θ, sin θ) | tan θ |
|---|---|---|
| 0° | (1, 0) | 0 |
| 30° | (√3/2, 1/2) | √3/3 |
| 45° | (√2/2, √2/2) | 1 |
| 60° | (1/2, √3/2) | √3 |
| 90° | (0, 1) | no value |
Pattern check
Sin θ down the table: √0/2, √1/2, √2/2, √3/2, √4/2. Cos θ is the same list upwards.
Unit circle signs in each quadrant
The axes cut the circle into 4 quadrants. Cos θ takes the sign of x, and sin θ takes the sign of y.
| Quadrant | Angles | cos θ | sin θ | tan θ | Positive |
|---|---|---|---|---|---|
| I | 0° to 90° | + | + | + | All |
| II | 90° to 180° | − | + | − | Sin |
| III | 180° to 270° | − | − | + | Tan |
| IV | 270° to 360° | + | − | − | Cos |
The last column reads All, Sin, Tan, Cos: “All Students Take Care”. Picture the point. In quadrant III, x and y are both negative, so tan θ = y ÷ x is positive.
Use the reference angle
The reference angle is the angle back to the x-axis: 180° − θ in quadrant II, θ − 180° in III, 360° − θ in IV. It gives the size; the quadrant gives the sign.
How to use the unit circle: 3 worked examples
Example 1: find sin 150°
150° is between 90° and 180°, so it is in quadrant II.
Reference angle: 180° − 150° = 30°. And sin 30° = 1/2.
y is positive in quadrant II. So sin 150° = 1/2.
Example 2: find cos 225°
225° is in quadrant III. Reference angle: 225° − 180° = 45°.
cos 45° = √2/2, but x is negative here.
So cos 225° = −√2/2, about −0.707.
Example 3: solve sin θ = 1/2 for 0° ≤ θ ≤ 360°
From the chart, sin 30° = 1/2.
Sin is also positive in quadrant II: 180° − 30° = 150°.
So θ = 30° or θ = 150°. Sin is negative in III and IV.
Common unit circle mistakes
- Swapping sin and cos. The point is (cos θ, sin θ), like (x, y). So cos 60° = 1/2, not √3/2.
- Wrong sign in quadrant II or III. cos 120° = −1/2, as the point is left of the y-axis.
- Measuring from the y-axis. Angles start on the positive x-axis. 30° from the y-axis is 60° from the x-axis.
- Copying √3/2 as 3/2. No coordinate on the unit circle is more than 1.
- Calculator in radian mode. In radians, sin 30 gives about −0.988. Set degrees first.
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- Swaps sin and cos: CNC, concept not clear.
- Wrong sign in quadrant II or III: CNC, concept not clear.
- Reads the angle from the wrong axis: DRC, did not read carefully.
- Copies √3/2 as 3/2: CWV, copied a wrong value.
- Calculator in radians: DRC, did not read carefully.
The Free BMI Maths Test finds which of the 6 mistake types your child makes, and their real maths level. About the Free BMI Maths Test
Test prep corner
The unit circle in SAT and JEE Main maths
The digital SAT reference sheet shows the 30°-60°-90° triangle (sides x, x√3, 2x) and the 45°-45°-90° triangle (sides s, s, s√2). It says a circle has 2π radians of arc. It gives no unit circle values. For JEE Main, Trigonometry is one of the 14 units in NTA’s maths topic list. These are SAT-style and JEE-style practice questions, not real past questions.
SAT-style: If sin(x°) = cos(26°) and 0 < x < 90, what is the value of x?
sin θ = cos(90° − θ). So x = 90 − 26 = 64. The points at 26° and 64° use the same two numbers, swapped.
JEE-style: Find every θ in [0, 2π] with 2cos²θ − 3cos θ + 1 = 0.
Factorise: (2cos θ − 1)(cos θ − 1) = 0. cos θ = 1/2 gives π/3 and 5π/3 (quadrants I and IV). cos θ = 1 gives 0 and 2π. So θ = 0, π/3, 5π/3, 2π. Trap: leaving out 2π.
Practise more with our SAT-style maths practice questions.
Facts checked on 11 November 2026 on satsuite.collegeboard.org and jeemain.nta.nic.in.
Quick check: 5 unit circle questions
1. What is the point on the unit circle at 60°?
Not quite. That is the 30° point.
Yes! cos 60° = 1/2 and sin 60° = √3/2.
Not quite. 3/2 is more than 1.
2. What is cos 120°?
Not quite. x is negative in quadrant II.
Not quite. The reference angle is 60°.
Yes! Reference angle 60°, negative x.
3. In which quadrant is sin θ negative and cos θ positive?
Not quite. That is sin positive, cos negative.
Yes! Bottom right: x positive, y negative.
Not quite. There, both are negative.
4. What is tan 225°?
Yes! x and y are both −√2/2, so y ÷ x = 1.
Not quite. Tan is positive here.
Not quite. That is the size of sin 45°.
5. What is sin 330°?
Yes! Reference angle 30°, negative y.
Not quite. Below the x-axis, sin is negative.
Not quite. That is cos 330°.
Got 2 or fewer right? Book the Free BMI Maths Test to find the cause.
Questions people ask about the unit circle
What is the unit circle?
The circle x² + y² = 1, with radius 1 and centre (0, 0).
Why is the radius 1?
With a long side of 1, cos θ is the x value and sin θ is the y value. No dividing.
How do I remember the unit circle?
Draw the 45° triangle and the half equilateral triangle. Then use reference angles.
Which values are negative in each quadrant?
Quadrant II: cos and tan. Quadrant III: sin and cos. Quadrant IV: sin and tan.
Is the unit circle on the SAT?
College Board lists “right triangles and trigonometry” and “circles” as SAT maths skills. The reference sheet gives the special right triangles, not unit circle values.
Keep exploring
Does your child mix up sin and cos?
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