TL;DR:
On the unit circle, the point at angle θ is
(cos θ, sin θ).Two triangles, 30-60-90 and 45-45-90, give every first-quadrant value.
Reflection and signs by quadrant give the rest, so there is almost nothing to memorize.
Many students are handed a unit circle chart with sixteen angles and told to memorize it. You can instead rebuild it in a minute from two triangles you can draw. The payoff is that you can also rebuild it when you forget.
What the unit circle says
The unit circle has radius 1 and centre at the origin. A point at angle θ, measured counter-clockwise from the positive x axis, has coordinates (cos θ, sin θ). Since the radius is 1, the hypotenuse of the little right triangle is 1, so cosine and sine are simply the x and y coordinates. By the Pythagorean theorem, cos² θ + sin² θ = 1.
Worked example: sin 30° = 1/2 from an equilateral triangle
Draw an equilateral triangle with side 2. Every angle is 60°.
Cut it in half with a vertical line from the top to the base. The base splits into 1 and 1.
The half-triangle is a right triangle with a 30° angle at the top, hypotenuse 2 and short leg 1.
The long leg, by Pythagoras, is
√(2² − 1²) = √3.The short leg 1 is opposite 30°. So
sin 30° = 1/2,cos 30° = √3/2andtan 30° = 1/√3.
The 60° angle flips the roles: sin 60° = √3/2, cos 60° = 1/2. Check with sin² + cos²: for 30°, 1/4 + 3/4 = 1. Correct.
The second triangle: 45-45-90
A square with side 1 has a diagonal of √2. Cut it along the diagonal and you get a right triangle with legs 1, 1 and hypotenuse √2. So sin 45° = 1/√2 = √2/2 and cos 45° = √2/2. Check: (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1.
Angle | sin | cos |
|---|---|---|
0° | 0 | 1 |
30° | 1/2 | √3/2 |
45° | √2/2 | √2/2 |
60° | √3/2 | 1/2 |
90° | 1 | 0 |
A memory aid for the sine column: it is √0/2, √1/2, √2/2, √3/2, √4/2, which simplifies to 0, 1/2, √2/2, √3/2, 1. Cosine is the same list backwards.
The rest of the circle by reflection
Angles in the other quadrants are reflections of a first-quadrant angle, called the reference angle. For 150°, the reference angle is 180° − 150° = 30°. Reflecting across the y axis keeps the height and flips the sign of the x coordinate, so sin 150° = 1/2 and cos 150° = −√3/2. For 210°, the reference angle is 210° − 180° = 30°, and both coordinates are negative: sin 210° = −1/2 and cos 210° = −√3/2.
Signs by quadrant
Quadrant I (0° to 90°): cosine and sine are both positive.
Quadrant II (90° to 180°): sine positive, cosine negative.
Quadrant III (180° to 270°): both negative.
Quadrant IV (270° to 360°): cosine positive, sine negative.
Try two values yourself
Find cos 135° and sin 300°. For 135°, the reference angle is 180° − 135° = 45°, in quadrant II where cosine is negative, so cos 135° = −√2/2. For 300°, the reference angle is 360° − 300° = 60°, in quadrant IV where sine is negative, so sin 300° = −√3/2.
Check each with the identity. For 135°, the sine is √2/2, and (−√2/2)² + (√2/2)² = 1/2 + 1/2 = 1. For 300°, the cosine is 1/2, and 3/4 + 1/4 = 1. A last one: tan 225°. The reference angle is 45° and both coordinates are negative, so tan 225° = (−√2/2) / (−√2/2) = 1.
Reading tangent from the same table
Tangent is sine divided by cosine, so the table gives it for free. At 30° it is (1/2) / (√3/2) = 1/√3 ≈ 0.577. At 45° it is (√2/2) / (√2/2) = 1. At 60° it is (√3/2) / (1/2) = √3 ≈ 1.732. Tangent grows without limit as the angle approaches 90°, because the cosine in the denominator shrinks toward zero. This is a good sanity check: a steep angle should give a large tangent.
How to check your answer
Pythagorean identity. Square your cosine and sine and add. You must get 1.
Size. Both values must lie between −1 and 1.
Picture. For 150°, the point is up and to the left, so sine is positive and cosine is negative.
Reference angle. The size of the answer must match the first-quadrant value.
Common mistakes
Swapping sine and cosine. Remember that the point is
(cos, sin), x first.Forgetting the sign in quadrants II to IV.
Using the angle itself instead of the reference angle.
Writing
sin 60° = 1/2. At 60° the height is large, so sine is the bigger one,√3/2 ≈ 0.866.Mixing 1/√3 with √3/2.
Where you will use it next
The next step is a better measure of angle than degrees, covered in radians: why 180 degrees equals π, which turns these angles into 30° = π/6, 45° = π/4 and so on. Everything here builds on the right-triangle ratios from SOH CAH TOA and when to use each ratio.
Frequently asked questions
Why is tangent not on the circle as a coordinate?
Tangent is the ratio sin θ / cos θ. It is the slope of the line from the origin to the point, and it is undefined where cos θ = 0, at 90° and 270°.
Do I still need to memorize anything?
Memorize the two triangles, or better, draw them. Then every other value is a reflection and a sign.
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