Calculating Height Using Trigonometry | IGCSE Maths

Calculating height using trigonometry is a core Edexcel IGCSE Maths skill: in a right-angled triangle you use SOHCAHTOA (sine, cosine and tangent) to find an unknown side or angle. On this page you will learn how to find the height of an object from a known distance, how to use angles of elevation and depression, and how to find a missing angle using inverse trigonometry. Every ratio is covered with step-by-step worked examples, followed by randomly generated, auto-marked practice questions below.

Prior Knowledge You should already be confident with Right-Angled Trigonometry (SOHCAHTOA), Pythagoras' Theorem and basic angle facts before starting this page.
⇩ Jump to Practice Questions ⇩

The Three Trigonometric Ratios (SOHCAHTOA)

In any right-angled triangle, the three sides are named relative to the angle \(\theta\) you are working with. The side opposite the right angle is always the hypotenuse. The side opposite \(\theta\) is the opposite, and the side next to \(\theta\) is the adjacent.

θ Opposite (O) Adjacent (A) Hypotenuse (H)

The hypotenuse is always opposite the right angle. Opposite and adjacent are named relative to the angle \(\theta\).

SOH
\[\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}\]
Sine = Opposite over Hypotenuse
CAH
\[\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}\]
Cosine = Adjacent over Hypotenuse
TOA
\[\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}\]
Tangent = Opposite over Adjacent

How to Choose the Right Ratio

  1. Label the sides: mark the hypotenuse, then label the opposite and adjacent relative to the given angle.
  2. Identify what you know and what you want: which two sides are involved?
  3. Pick the ratio: the ratio that uses both those sides is the one to use.
  4. Set up the equation and solve for the unknown.

Quick guide: know/want Opp & Hyp β†’ use sin. Know/want Adj & Hyp β†’ use cos. Know/want Opp & Adj β†’ use tan.

Angles of Elevation and Depression

θ angle of elevation Observer Object

Angle of elevation: measured upwards from the horizontal to the line of sight.

Obser- ver θ angle of depression Object

Angle of depression: measured downwards from the horizontal to the line of sight.

Core Ideas

Label first Always label hypotenuse, opposite and adjacent before picking a ratio.
Which ratio? The ratio you need involves the two sides you know or want. SOH, CAH, or TOA.
Rearrange carefully If the unknown is on the bottom of the fraction, multiply both sides. If on top, divide.
Finding an angle Use inverse trig: \(\theta = \sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\) on your calculator.

Finding a Side

πŸ’‘ Using Sine: Height of a Drone

A drone hovers at the end of a taut 24 m tether fixed to a peg on level ground. The tether makes an angle of elevation of 38Β° with the ground. Find the height \(h\) of the drone above the ground.

38Β° adjacent h = ? 24 m (hyp)
1
Identify sidesWant: Opposite (height \(h\)). Have: Hypotenuse (24 m). Use SOH.
2
Set up and solve
\[\sin 38Β° = \frac{h}{24}\]
\[h = 24 \times \sin 38Β° = 14.8 \text{ m (3 s.f.)}\]

πŸ’‘ Using Tangent: Tower Height

A 25 m tower PQ stands on level ground. An engineer's mark R is on the ground. The angle of elevation of the top from R is 60Β°. Find the distance RQ.

60Β° RQ = p (adj) 25 m (opp) hyp
1
Identify sidesWant: Adjacent (\(p\)). Have: Opposite (25 m). Use TOA.
2
Set up and solve
\[\tan 60Β° = \frac{25}{p}\]
\[p = \frac{25}{\tan 60Β°} = 14.4 \text{ m (3 s.f.)}\]

πŸ’‘ Using Cosine: Finding the Hypotenuse

A mooring rope runs from a boat down to an anchor block on the seabed. The rope makes a 35Β° angle with the horizontal water surface, and the horizontal distance from the boat to the block is 18 m. Find the length \(L\) of the rope.

1
Identify sidesWant: Hypotenuse (\(L\)). Have: Adjacent (18 m). Use CAH.
2
Set up the equation
\[\cos 35Β° = \frac{18}{L}\]
3
Rearrange and solve
\[L = \frac{18}{\cos 35Β°} = 22.0 \text{ m (3 s.f.)}\]

πŸ’‘ Angle of Depression: Distance on Ground

From a window 10 m above the ground, the angle of depression to a coin on the ground is 15Β°. Find the horizontal distance from the coin to the base of the building.

1
Draw the triangleThe angle of depression from the window equals the angle of elevation from the coin (alternate angles). Height = 10 m (opposite), distance = \(d\) (adjacent). Use TOA.
2
Set up and solve
\[\tan 15Β° = \frac{10}{d}\]
\[d = \frac{10}{\tan 15Β°} = 37.3 \text{ m (3 s.f.)}\]

Finding an Angle

When you know two sides, use the inverse trig functions on your calculator: \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).

πŸ’‘ Finding an Angle of Elevation

A cliff is 87 m tall. You stand 150 m from its base on level ground. Find the angle of elevation to the top of the cliff.

1
Identify sidesOpposite = 87 m (height). Adjacent = 150 m (distance). Use TOA with inverse tan.
2
Set up and solve
\[\tan\theta = \frac{87}{150}\]
\[\theta = \tan^{-1}\!\left(\frac{87}{150}\right) = 30.1Β° \text{ (3 s.f.)}\]

πŸ’‘ Finding an Angle: Ladder Against a Wall

A 5 m ladder leans against a vertical wall. Its top is 3 m above the ground. Find the angle the ladder makes with the ground.

1
Identify sidesOpposite = 3 m (height up wall). Hypotenuse = 5 m (ladder). Use SOH with inverse sin.
2
Set up and solve
\[\sin\theta = \frac{3}{5} = 0.6\]
\[\theta = \sin^{-1}(0.6) = 36.9Β° \text{ (3 s.f.)}\]

πŸ”‘ Key Points

  • Label hypotenuse, opposite, and adjacent before choosing a ratio.
  • SOH: sin = Opp/Hyp. CAH: cos = Adj/Hyp. TOA: tan = Opp/Adj.
  • Angle of elevation is measured up from the horizontal.
  • Angle of depression is measured down from the horizontal.
  • To find an angle, use \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).
  • Always give answers to 3 significant figures unless told otherwise.

⚠️ Common Pitfalls

  • Using the wrong ratio: always label sides first.
  • Dividing the wrong way round when rearranging.
  • Confusing the angle of depression with the angle inside the triangle: they are equal (alternate angles), but make sure you draw the diagram.
  • Calculator not in degree mode.
  • Using sin/cos/tan instead of the inverse when finding an angle.
⇩ Jump to Practice Questions ⇩

Ready to practise? Work through the rooms below.

Trigonometry: Practice Room

Practise finding sides and angles in right-angled triangles using SOHCAHTOA. Give lengths to 3 significant figures and angles to 1 decimal place.

βœ“ Correct 0
βœ— Re-attempts 0
πŸ”₯ Streak 0
πŸ† Best 0

Lengths: 3 significant figures  |  Angles: 1 decimal place (e.g. 34.2)

Video guide on how to calculate height using Trigonometry

This YouTube video provides a detailed explanation of using trigonometry to calculate the height of objects. It covers the SOHCAHTOA method, explains the trigonometric ratios, and demonstrates how to apply them in practical scenarios. By watching the video, you will gain a clear understanding of using trigonometry for height calculations and be able to solve problems with confidence.