For an angle above 90°, find its acute reference angle, take the ratio of that angle, and then attach the sign that the quadrant allows. The size comes from the reference angle and the sign comes from the quadrant.
This lesson opens the trigonometric ratios and graphs chapter. The next lesson, reading sine, cosine and tangent graphs, shows the same signs on a graph.
How do I find the reference angle?
The reference angle is the acute angle to the horizontal axis. Use the rule for the quadrant the angle sits in.
| Quadrant | Angle range | Reference angle | Positive ratios |
|---|---|---|---|
| 1 | 0° to 90° | θ | sin, cos, tan |
| 2 | 90° to 180° | 180° − θ | sin |
| 3 | 180° to 270° | θ − 180° | tan |
| 4 | 270° to 360° | 360° − θ | cos |
Worked example: the ratios of 150°
An angle of 150° is in quadrant 2, so the reference angle is 180° − 150° = 30°.
- sin 150° = +sin 30° = 0.5, because sine is positive in quadrant 2.
- cos 150° = −cos 30° = −0.866, because cosine is negative there.
- tan 150° = −tan 30° = −0.577, because tangent is negative there.
All three use the same reference angle, 30°. The quadrant table decides each sign.
The mistake that costs marks
The common slip is to use the reference angle and forget the sign. A student writes cos 150° = 0.866 because cos 30° = 0.866.
| Step | Wrong | Right |
|---|---|---|
| Locate the angle | (skipped) | 150° is in quadrant 2 |
| Reference angle | 30° | 30° |
| Sign of cosine | positive by default | negative in quadrant 2 |
| Answer | 0.866 | −0.866 |
Before you write any answer, name the quadrant aloud. That single line prevents most sign errors.
Check yourself
Find cos 240° and sin 300°, without a calculator, using values of the 30° and 60° angles.
Answer
240° is in quadrant 3, so the reference angle is 240° − 180° = 60°. Cosine is negative in quadrant 3, so cos 240° = −cos 60° = −0.5.
300° is in quadrant 4, so the reference angle is 360° − 300° = 60°. Sine is negative in quadrant 4, so sin 300° = −sin 60° = −0.866.
What to study next
The signs you just practised appear as the height of a wave in the next lesson. Move on to reading sine, cosine and tangent graphs, then try the trigonometry practice set. If a slip starts in the algebra, the algebra step repair trainer targets that step.
For a teacher to go through these signs with you, see online one-to-one Mathematics tuition.