Right-Angled Trigonometry

What right-angled trigonometry does

Right-angled trigonometry connects an acute angle in a right-angled triangle with the lengths of its sides. It lets you find a missing side when you know one side and an acute angle, or find a missing angle when you know two sides.

The connections work because right-angled triangles with the same acute angles are similar. Enlarging such a triangle changes all its side lengths by the same scale factor, so the ratios between corresponding sides remain constant.

Label the triangle relative to the angle

First identify the right angle. The side opposite it is the hypotenuse, which is always the longest side.

Next choose the acute angle $\theta$ involved in the question. The side directly across from $\theta$ is the opposite side. The remaining side is the adjacent side: it touches $\theta$ but is not the hypotenuse.

The hypotenuse stays the same whichever acute angle is selected, but the opposite and adjacent sides exchange roles when the selected angle changes. This is why sides must be labelled for the particular angle in the question rather than from the triangle's orientation on the page.

The three trigonometric ratios

The required ratios are

$$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\qquad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}.$$

The mnemonic SOH CAH TOA records these relationships:

• SOH: sine = opposite ÷ hypotenuse;
• CAH: cosine = adjacent ÷ hypotenuse;
• TOA: tangent = opposite ÷ adjacent.

Choose the ratio containing both the side you know and the side you need. The third side should not appear in the equation.

Finding a missing side

Suppose the hypotenuse is $7\text{ cm}$, an acute angle is $32^\circ$, and the side opposite that angle is $x$. Opposite and hypotenuse indicate sine:

$$\sin 32^\circ=\frac{x}{7}.$$

Multiplying both sides by $7$ gives

$$x=7\sin 32^\circ\approx3.7\text{ cm}.$$

Sometimes the unknown is in the denominator. If an adjacent side is $4.2\text{ cm}$, the hypotenuse is $y$, and the included acute angle is $64^\circ$, use cosine:

$$\cos64^\circ=\frac{4.2}{y}.$$

Rearranging gives

$$y=\frac{4.2}{\cos64^\circ}\approx9.6\text{ cm}.$$

This is reasonable because a hypotenuse must be longer than either of the other sides. A quick check like this can expose an inverted fraction or an incorrect rearrangement.

Finding a missing angle

When two side lengths are known, first choose the ratio connecting them. For example, if the opposite side is $3\text{ m}$ and the adjacent side is $7\text{ m}$, then

$$\tan\theta=\frac{3}{7}.$$

Use the inverse tangent operation:

$$\theta=\tan^{-1}\left(\frac{3}{7}\right)\approx23.2^\circ.$$

Similarly, use $\sin^{-1}$ with an opposite-to-hypotenuse ratio and $\cos^{-1}$ with an adjacent-to-hypotenuse ratio. Here, the superscript $$ means the inverse function that recovers the angle; it does not mean .

A dependable method

1. Confirm that the triangle is right-angled.
2. Mark the acute angle being used.
3. Label the sides H, O and A relative to that angle.
4. Identify the known side and the required side or angle.
5. Select sine, cosine or tangent so that the equation contains those two sides.
6. Substitute, rearrange and calculate in degree mode.
7. Round only at the end, give the requested accuracy and attach a length unit to a side or degrees to an angle.

Trigonometric ratios themselves have no units because each is one length divided by another length. The missing length, however, must be reported in the units used in the question.

Label first

• H: opposite the right angle and the longest side.
• O: opposite the selected acute angle.
• A: beside the selected angle, excluding the hypotenuse.
• Changing the selected angle swaps O and A, not H.

Recall the ratios

$$\sin\theta=\frac{O}{H},\qquad \cos\theta=\frac{A}{H},\qquad \tan\theta=\frac{O}{A}.$$