Notable values and properties
Notable angles and symmetries on a circle
Notable angles
Certain angles crop up very frequently in trigonometry because their cosine and sine values are ‘simple’ (often involving square roots). These are 0, π/6 (30 degrees), π/4 (45 degrees), π/3 (60 degrees) and π/2 (90 degrees).
Table of notable values
| Angle (rad) | 0 | π/6 | π/4 | π/3 | π/2 |
|---|---|---|---|---|---|
| Angle (deg) | 0 | 30 | 45 | 60 | 90 |
| cos | 1 | sqrt(3)/2 | sqrt(2)/2 | 1/2 | 0 |
| sin | 0 | 1/2 | sqrt(2)/2 | sqrt(3)/2 | 1 |
Tip: the same values (1/2, sqrt(2)/2, sqrt(3)/2) appear in reverse order for cos and sin, which helps you to memorise them.
Symmetries on the unit circle
The unit circle exhibits symmetries that allow us to quickly find the cosine and sine of angles related to a known angle θ:
| Angle | cos | sin |
|---|---|---|
| -theta (symmetry / horizontal axis) | cos(theta) | -sin(theta) |
| pi - theta (symmetry / vertical axis) | -cos(theta) | sin(theta) |
| π + θ (central symmetry) | −cos(θ) | −sin(θ) |
| π/2 − θ (symmetry about the bisector) | sin(θ) | cos(θ) |
Example
Calculate cos(2 × π/3). Note that 2 × π/3 = π – π/3, so cos(2 × π/3) = –cos(π/3) = –1/2.
Common pitfall
Be careful with the sign: symmetry about the vertical axis (π – θ) changes the sign of the cosine but not that of the sine. Always make sure you identify which axis of symmetry is being used before applying the rule.