What Are the Exact Trigonometric Values for 0; π/6; π/4; π/3; and π/2?

The exact values of sine, cosine, and tangent for the five key first-quadrant angles are shown below. These values must be recalled and used without a calculator in IB Mathematics: Analysis and Approaches, especially on Paper 1.

Why These Values Work

On the unit circle, the point corresponding to an angle $\theta$ has coordinates

$$(\cos\theta,\sin\theta).$$

Therefore, cosine is the horizontal coordinate and sine is the vertical coordinate. As $\theta$ increases from $0$ to $\frac{\pi}{2}$, sine increases from $0$ to , while cosine decreases from to .

A useful pattern is

$$\sin\theta=\frac{\sqrt{0}}{2},\frac{\sqrt{1}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2},\frac{\sqrt{4}}{2}.$$

Cosine uses the same list in reverse order. Then calculate tangent using

$$\tan\theta=\frac{\sin\theta}{\cos\theta}.$$

This pattern is a memory aid, not a substitute for understanding. It works because the special-triangle coordinates lie on the unit circle, where each point satisfies $x^2+y^2=1$. Knowing the geometric origin helps you reconstruct a forgotten value reliably.

These values can also be derived from two special triangles. Splitting an equilateral triangle of side length $2$ creates a $30^\circ$-$60^\circ$-$90^\circ$ triangle with side lengths , , and . The side ratios give the values for and . A right isosceles triangle with legs and hypotenuse gives the values for .

For example,

$$\tan\left(\frac{\pi}{6}\right)=\frac{1/2}{\sqrt{3}/2}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}.$$

At $\theta=\frac{\pi}{2}$, cosine is zero, so tangent requires division by zero and is undefined. A common misconception is that $\tan\left(\frac{\pi}{2}\right)=0$; it is undefined, not zero.

For angles outside the first quadrant, keep the same reference-angle magnitudes and change only the signs. For example, $\frac{5\pi}{6}$ has reference angle $\frac{\pi}{6}$. In quadrant II, sine is positive while cosine and tangent are negative, so . This avoids memorizing a second table.

IB Exam Technique

This is SL 3.5: Unit circle definitions and exact trigonometric ratios. On Paper 1, show exact fractions and surds rather than decimals. For other quadrants, use the reference angle and determine the sign from the unit circle. Radians are assumed unless degrees are specified.