Exact Trigonometric Values

For some angles, sine, cosine and tangent can be written exactly using fractions and square roots. For example, $\sin 60^\circ=\frac{\sqrt{3}}{2}$. A decimal such as $0.866$ is only an approximation: the surd expression retains the full value.

Two right-angled triangles provide the exact values for $30^\circ$, $45^\circ$ and $60^\circ$. Their usefulness comes from similarity: all triangles with the same angles have the same side ratios, even when their sizes differ.

For a chosen acute angle $\theta$, the opposite side is across from the angle, the adjacent side is next to it but is not the hypotenuse, and the hypotenuse is opposite the right angle. The ratios are:

These are remembered as SOHCAHTOA. Since each is a ratio of two lengths, their units cancel.

Building the 30°–60°–90° triangle

Start with an equilateral triangle whose sides are each $2$ units long. Every angle is $60^\circ$. Draw the perpendicular from the top vertex to the midpoint of the base. This line of symmetry splits the triangle into two identical right-angled triangles.

Each half has a hypotenuse of $2$ units and a base of $1$ unit. The top angle is halved to $30^\circ$, while the angle at the original base remains $60^\circ$.

Use Pythagoras’ theorem to find the height $h$:

So the sides opposite $30^\circ$, $60^\circ$ and $90^\circ$ are in the ratio:

Building the 45°–45°–90° triangle

Now take a right-angled isosceles triangle with both shorter sides $1$ unit long. The two remaining angles are equal and add to $90^\circ$, so each is $45^\circ$.

Pythagoras’ theorem gives its hypotenuse $c$:

The side ratio is therefore:

Reading exact trigonometric values from the sides

In the $30^\circ$–$60^\circ$–$90^\circ$ triangle, choose the $30^\circ$ angle first. Its opposite side is $1$, its adjacent side is $\sqrt{3}$ and its hypotenuse is $2$. Therefore:

The final tangent form comes from multiplying the numerator and denominator by $\sqrt3$: the denominator becomes $3$.

When you choose the $60^\circ$ angle instead, opposite and adjacent swap roles. The hypotenuse remains $2$, giving:

For either $45^\circ$ angle in the isosceles triangle, opposite and adjacent are both $1$. Hence:

Sine and cosine are equal at $45^\circ$ because the two shorter sides are equal. For $30^\circ$ and $60^\circ$, the sine values match the cosine values in reverse order because the opposite and adjacent sides exchange roles.

Using the ratios at any size

The numbers in these ratios are not fixed side lengths. Multiplying every side by the same positive scale factor $k$ gives:

• $30^\circ$–$60^\circ$–$90^\circ$: $k$, $k\sqrt3$, $2k$.
• $45^\circ$–$45^\circ$–$90^\circ$: $k$, , .

For example, suppose a $30^\circ$–$60^\circ$–$90^\circ$ triangle has a hypotenuse of $10\text{ cm}$. Since $2k=10$, the scale factor is $k=5$. Its shorter side is $5\text{ cm}$ and its longer side is $5\sqrt3\text{ cm}$.

The same longer side can be found using $\sin60^\circ=\frac{\sqrt3}{2}$: multiplying this ratio by the $10\text{ cm}$ hypotenuse gives $5\sqrt3\text{ cm}$. The side-ratio method and the trigonometric method are two ways of using the same relationship.