Method · Trigonometry
How to Find Sine, Cosine and Tangent of an Angle
Every angle reduces to the same four moves: shrink it to one turn, find its quadrant, take the reference angle, then attach the sign.
The four moves
To learn how to find sine, cosine and tangent of any angle, the method is the same each time.
- Reduce the angle to between 0° and 360° by adding or subtracting whole turns of 360°. The result is a coterminal angle.
- Find the quadrant, which says which functions are positive.
- Find the reference angle: the acute angle between the terminal side and the x-axis.
- Read the value for the reference angle from the standard table, then attach the sign.
The standard values to know
Only five reference angles are needed. Everything else is built from them.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Notice that sine counts upward through the table while cosine counts the same values downward. That is not luck: sin θ = cos (90° − θ).
An angle inside the first quadrant
- 60°already between 0° and 90°, so it is its own reference angle
- sin 60° = √3/2read straight from the table, positive in quadrant I
An angle in another quadrant
- 135°between 90° and 180°, so quadrant II
- 180° − 135° = 45°the reference angle
- cos 45° = √2/2the size of the answer
- cos 135° = −√2/2cosine is negative in quadrant II
- 210°between 180° and 270°, so quadrant III
- 210° − 180° = 30°the reference angle
- tan 30° = √3/3the size of the answer
- tan 210° = √3/3tangent is positive in quadrant III
Negative angles and large angles
- 750° − 2 × 360° = 30°remove two whole turns to find the coterminal angle
- sin 30° = 1/2the reference angle is 30° in quadrant I
- sin 750° = 1/2a full turn does not change the value
- −30° + 360° = 330°add a turn to bring it into 0° to 360°
- 360° − 330° = 30°the reference angle, in quadrant IV
- sin (−30°) = −1/2sine is negative in quadrant IV
Angles on an axis, and undefined values
An angle of 90°, 180° or 270° sits on an axis, so it has no quadrant. Read it straight off the circle instead: at 90° the point is (0, 1), so sin = 1, cos = 0 and tan = 1 ÷ 0, which is undefined. An undefined value is not zero and not infinity: the division it needs has no answer.
Radians, and checking your answer
The same method works in radians: π/4 is 45°, 3π/4 is 135°, and so on. Make sure your calculator mode matches the unit, since sin 30 is 0.5 in degrees and −0.988 in radians.
To check your answer, test the sign against the quadrant, and test the size against the reference angle's value. The Trig Calculator on this site follows the same four moves and shows the quadrant and reference angle at each step.
Questions about how to find sine cosine and tangent
What is sin 135°?
√2/2. The reference angle is 45° and sine is positive in the second quadrant.
What is cos 135°?
−√2/2. The reference angle is 45° and cosine is negative in the second quadrant.
What is tan 90°?
Undefined. Tangent is sine over cosine, and cosine is 0 at 90°, so the division has no answer.
How do I find the trig value of a negative angle?
Add 360° until it falls between 0° and 360°, then use the quadrant and reference angle. sin(−30°) becomes sin 330°, which is −1/2.