IP EMaths Notes (Upper Sec, Year 3-4): 09) Right-Triangle Trigonometry

Right-angled triangles provide fast access to heights, bearings, and resultant forces. Remember which side is opposite, adjacent, or hypotenuse relative to the marked angle.

Ratio recap

• \(\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\).
• \(\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}\).
• \(\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}\).
• Pythagoras: \(c^{2} = a^{2} + b^{2}\) for hypotenuse \(c\).

Worked example - Ladder against a wall

A ladder leans against a vertical wall, touching the ground 3.6 metres from the wall. The ladder makes an angle of 68 degrees with the ground. Find the length of the ladder and the height it reaches on the wall.

1. Let \( L \) denote the ladder length (hypotenuse). The adjacent side measures \( \pu{3.6 m} \).
2. Use the cosine ratio:
   \[ \cos\bigl(68^\circ\bigr) = \frac{\pu{3.6 m}}{L} \quad\Rightarrow\quad L = \frac{\pu{3.6 m}}{\cos\bigl(68^\circ\bigr)} \approx \pu{9.61 m}. \]
3. Height on the wall via sine:
   \[ h = L \times \sin\bigl(68^\circ\bigr) \approx \pu{9.61 m} \times 0.9272 \approx \pu{8.91 m}. \]

Try this

A surveyor stands \( \pu{45 m} \) from the base of a building. The angle of elevation to the roof is \( 32^\circ \). Estimate the building height to the nearest tenth of a metre.