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Right Triangle Trigonometry

General Mathematics

About these notes

Area Of Study

Geometry and measurement

Unit

Unit 3: General Mathematics Unit 3

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Right Triangle Trigonometry

General Mathematics
01 May 2026

Right-Angled Triangle Trigonometry

The Three Ratios

For a right-angled triangle with angle \(\theta\):

\[\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}\]

Mnemonic: SOH–CAH–TOA

Ratio Use when you have…
\(\sin\) opposite and hypotenuse
\(\cos\) adjacent and hypotenuse
\(\tan\) opposite and adjacent

Labelling the Triangle

The sides depend on which angle \(\theta\) you are using:
- Hypotenuse: opposite the right angle (always the longest side)
- Opposite: the side directly opposite angle \(\theta\)
- Adjacent: the side next to angle \(\theta\) (not the hypotenuse)

Finding a Side

\[x = \text{hypotenuse} \times \sin\theta \quad \text{or} \quad x = \text{hypotenuse} \times \cos\theta \quad \text{etc.}\]

Worked Example

In a right-angled triangle, the hypotenuse is 15 m and one angle is \(35°\). Find the opposite side.

\[\sin(35°) = \frac{\text{opposite}}{15} \implies \text{opposite} = 15\sin(35°) \approx 15 \times 0.5736 \approx 8.60 \text{ m}\]

Finding an Angle

Use the inverse trigonometric functions: \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\).

\[\theta = \sin^{-1}\!\left(\frac{\text{opp}}{\text{hyp}}\right)\]

Worked Example

A ramp rises 1.2 m over a horizontal distance of 5.5 m. Find the angle of inclination.

\[\tan\theta = \frac{1.2}{5.5} \implies \theta = \tan^{-1}\!\left(\frac{1.2}{5.5}\right) \approx \tan^{-1}(0.2182) \approx 12.3°\]

Angle of Elevation and Depression

  • Angle of elevation: angle measured upward from horizontal to an object above
  • Angle of depression: angle measured downward from horizontal to an object below

Both are measured from the horizontal, not the vertical.

Worked Example

From a cliff 80 m high, the angle of depression to a boat at sea is \(22°\). Find the horizontal distance to the boat.

\[\tan(22°) = \frac{80}{d} \implies d = \frac{80}{\tan(22°)} \approx \frac{80}{0.4040} \approx 198 \text{ m}\]

Bearings

Bearings are measured clockwise from North and written as three digits, e.g., 045°, 270°.

REMEMBER: Always draw a diagram. Label the right angle, the known angle \(\theta\), and the known side. Then select the correct ratio (SOH-CAH-TOA).

EXAM TIP: Set your CAS to degree mode for all trigonometry problems in General Mathematics. Check: \(\sin(30°)\) should give 0.5.

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