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Right-Angled Trigonometry

In any right-angled triangle, the angles and the sides are locked together by three ratios. Learn SOHCAHTOA and you can find a missing side or a missing angle from just two known pieces.

MYP 4Extended MathsTrigonometryCriteria A · C · D~11 min read

Written by Fayek Jawad Haider · MYP student

Labelling the sides

Trigonometry only works once the sides are named relative to the angle you are using. Pick the angle in question (never the right angle) and label from there.

Hypotenuse
The longest side, always opposite the right angle. It never changes, whichever angle you focus on.
Opposite
The side directly across from the angle you are using.
Adjacent
The side next to the angle you are using that is not the hypotenuse.

SOHCAHTOA

The three ratios are captured by the mnemonic SOHCAHTOA, which packs the three definitions into one word:

\[ \sin\theta = \frac{\text{O}}{\text{H}}, \qquad \cos\theta = \frac{\text{A}}{\text{H}}, \qquad \tan\theta = \frac{\text{O}}{\text{A}} \]

Here O, A and H are the opposite, adjacent and hypotenuse, and \(\theta\) is the angle you chose. To decide which ratio to use, look at which two sides the question involves: the one that has both is the ratio you want.

Read the mnemonic in threes

SOH, CAH, TOA. Each block is ratio, top, bottom: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. Set your calculator to degrees before you start.

Finding a missing side

When you know an angle and one side and want another side, choose the ratio linking the two sides, then rearrange.

Worked example

A right-angled triangle has hypotenuse 12 cm and an angle of \(60^\circ\). Find the length of the side adjacent to that angle.

1
You have the hypotenuse and want the adjacent, so use cosine: \(\cos\theta = \dfrac{\text{A}}{\text{H}}\).
2
Substitute: \(\cos 60^\circ = \dfrac{\text{A}}{12}\).
3
Rearrange: \(\text{A} = 12 \times \cos 60^\circ = 12 \times 0.5\).
Adjacent \(= 6\) cm

Finding a missing angle

When you know two sides and want the angle, form the correct ratio, then undo it with the inverse function, written \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\) (on the calculator, usually the shift or 2nd key).

Worked example

A right-angled triangle has an opposite side of 6 cm and an adjacent side of 8 cm for angle \(\theta\). Find \(\theta\) to 1 decimal place.

1
You have the opposite and adjacent, so use tangent: \(\tan\theta = \dfrac{\text{O}}{\text{A}}\).
2
Substitute: \(\tan\theta = \dfrac{6}{8} = 0.75\).
3
Take the inverse: \(\theta = \tan^{-1}(0.75) = 36.86\ldots^\circ\).
\(\theta \approx 36.9^\circ\)

Where this is assessed

Selecting the correct ratio and computing the answer is Criterion A (Knowing and understanding). Setting out the ratio, the substitution and a labelled sketch is Criterion C (Communicating). Applying it to heights, ramps and navigation is Criterion D (Applying maths in context). If your angle comes out larger than \(90^\circ\) in a right-angled triangle, you have almost certainly used a side ratio the wrong way up.

Check yourself

1. The opposite side is 6 cm and the angle is \(30^\circ\). Find the hypotenuse. +

Opposite and hypotenuse means sine: \(\sin 30^\circ = \dfrac{6}{\text{H}}\). Rearranging, \(\text{H} = \dfrac{6}{\sin 30^\circ} = \dfrac{6}{0.5} = \mathbf{12}\) cm.

2. The adjacent side is 4 cm and the hypotenuse is 8 cm. Find the angle. +

Adjacent and hypotenuse means cosine: \(\cos\theta = \dfrac{4}{8} = 0.5\). So \(\theta = \cos^{-1}(0.5) = \mathbf{60^\circ}\).

3. Which ratio links the opposite and the adjacent sides? +

The ratio using opposite over adjacent is tangent (the TOA in SOHCAHTOA). It is the one to reach for when the hypotenuse is neither given nor wanted.

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