In trigonometry, a small set of angles appears constantly in problems, graphs, and identities. Their trigonometric ratios have exact values that can be written as simple fractions or surds, so you are expected to recall them instantly rather than approximate them with a calculator.

When angles are measured in radians, the three key angles are:

• $\dfrac{\pi}{6}$ (equivalent to $30^\circ$)
• $\dfrac{\pi}{4}$ (equivalent to $45^\circ$)
• $\dfrac{\pi}{3}$ (equivalent to $60^\circ$)

The exact values

Sine

Cosine

Tangent

Patterns that make recall easier

• Sine increases with the angle. As the angle grows from $\dfrac{\pi}{6}$ to $\dfrac{\pi}{3}$, the sine value rises from $\dfrac{1}{2}$ to $\dfrac{\sqrt{3}}{2}$.
• Cosine decreases with the angle. The cosine values are the sine values in reverse order: $\dfrac{\sqrt{3}}{2}$, then $\dfrac{\sqrt{2}}{2}$, then $\dfrac{1}{2}$.
• Sine and cosine are complementary. For example, $\sin\dfrac{\pi}{6} = \cos\dfrac{\pi}{3}$ and $\sin\dfrac{\pi}{3} = \cos\dfrac{\pi}{6}$.
• Tangent is sine divided by cosine. This means you only need to remember the sine and cosine values; tangent follows from $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
• The value at $\dfrac{\pi}{4}$ is symmetric. Both sine and cosine equal $\dfrac{\sqrt{2}}{2}$, so tangent equals $1$.

Rationalising denominators

Values such as $\dfrac{1}{\sqrt{2}}$ and $\dfrac{1}{\sqrt{3}}$ are often written with a rational denominator:

Both forms are correct, but the rationalised form is usually preferred in final answers.

Common pitfalls

• Mixing up the sine and cosine values for $\dfrac{\pi}{6}$ and $\dfrac{\pi}{3}$.
• Forgetting that $\tan\dfrac{\pi}{4} = 1$ exactly, not approximately.
• Writing $\dfrac{1}{\sqrt{2}}$ without rationalising when a simplified form is required.
• Confusing radian measures with degree measures; $\dfrac{\pi}{6}$ is not the same as $6^\circ$.