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Teach Some Applications of Trigonometry in five steps.

For each idea: show it, ask before you tell, then say one line.

Gather a ruler, books, paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

  • You look more steeply upward as you approach a tall building.
  • A tower’s shadow shortens as the Sun climbs higher.
  • A rooftop pole’s height can be found without reaching the roof.

What to teach

Build the right triangle

Line of sight, angle of elevation and angle of depressionSight angles start from the horizontal through your eyes.Height of an object above the observer’s eyeWhole height includes the section below your eyes.

Use two triangles

Height of a flagstaffSubtract the building’s height from the combined height.Height of a towerSame-side distances are subtracted.Distance between two objectsOpposite-side distances add to give the whole width.

How to teach it

Build the right triangle

A ruler lying level on a book stack in front of a doorway.

1. Line of sight, angle of elevation and angle of depression

Sight angles start from the horizontal through your eyes.

Show
Face a doorway with a ruler horizontal beside your eye. Sight its top, then bottom. Sketch sights and parallel horizontals.
Ask
Where will each angle start?
Say
Your line of sight joins your eye to the point you view. Measure elevation above your horizontal. Measure depression below it.
Used in
Ladder length and ground angle determine its reach to a window.
If your child says…“I measure the viewing angle from the vertical doorframe.”
Ask
Does your level ruler point along the floor or up the frame?
What’s true
Elevation and depression match when viewing the same line from opposite ends, using parallel horizontals.
A building’s height split into rise above eye level and eye height below.

2. Height of an object above the observer’s eye

Whole height includes the section below your eyes.

Show
Find the eye point in the diagram. Follow the horizontal to the building. Point to the sections above and below it.
Ask
Will adding the section below eye level give a height greater than the rise alone?
Say
It will, because the building also extends below eye level. Add eye height to find the whole height.
Used in
Ground measurements give tree heights without climbing.
If your child says…“The rise above my eyes is the whole height.”
Ask
Which section lies below the horizontal line?
What’s true
On level ground, the building extends both above and below your eyes.

Use two triangles

Two sight lines reach a building’s roof and flagstaff tip from one ground point.

3. Height of a flagstaff

Subtract the building’s height from the combined height.

Show
Find the building in the diagram. Trace both sight lines and shared base d. Flagstaff height f uses metres.
Ask
How can we find the flagstaff’s height alone?
Say
The 45° roof angle gives a 2 m base. Subtract the building’s height: f=2tan⁡60∘−2f=2\tan60^\circ-2.
Used in
A rooftop aerial’s height can be calculated from the ground, avoiding roof access.
If your child says…“The taller triangle gives the flagstaff’s height.”
Ask
Does the taller triangle start at the roof or at the ground?
What’s true
The taller triangle includes the building underneath the flagstaff.
Two shadow triangles share a tower; the longer shadow extends 2 m beyond the shorter.

4. Height of a tower

Same-side distances are subtracted.

Show
Find the shared tower height h, in metres, in the diagram. Trace both complete shadows from the tower’s foot. Find the extra 2 m beyond the shorter shadow.
Ask
Will raising the Sun’s angle from 30° to 60° shorten or lengthen this tower’s shadow?
Say
It shortens the shadow because the same height is reached more steeply. Subtract the shorter shadow from the longer: htan⁡30∘−htan⁡60∘=2\frac{h}{\tan30^\circ}-\frac{h}{\tan60^\circ}=2. Solving gives a tower height of root three metres.
Used in
Distance differences give travel distances for calculating speed.
If your child says…“The extra 2 m is the whole shadow.”
Ask
Where does each complete shadow start?
What’s true
The extra length starts where the shorter shadow ends, rather than at the tower.
A pole divides the distance between opposite endpoints into two adjoining horizontal bases.

5. Distance between two objects

Opposite-side distances add to give the whole width.

Show
Find the pole between the endpoints in the diagram. Trace both horizontal bases and the whole width. Call the left distance x and the whole width W, both in metres.
Ask
Will the whole width be greater than either horizontal base?
Say
Yes, because the bases meet end to end. Add their lengths: W=3tan⁡30∘+3tan⁡60∘W=\frac{\sqrt3}{\tan30^\circ}+\frac{\sqrt3}{\tan60^\circ}. The bases measure three metres and one metre, giving a total width of four metres.
Used in
Angles measured from a bridge give a river’s width without crossing the water.
If your child says…“I subtract the distances to get the width.”
Ask
Tracing from left to right, do you cross one stretch or both?
What’s true
The pole stands between the endpoints, so the width includes both horizontal stretches.

A real problem

Angled lights signalled the required flight height.

The Dambusters raid happened in 1943.

Barometric altimeters lacked the accuracy crews needed to determine altitude.

The light spots met on the water at the required release height.

WikipediaRAF Museum

Torches angled downward so their light spots overlap on a tabletop.

Concept Lab · Common misconception

The tempting mistake in Some Applications of Trigonometry

Open standalone misconception page →

Which height does an elevation angle measure?

Wrong

The height calculated from eye level is the whole tower height.

Why it feels right. I picture the triangle starting on the ground.

Right

Add eye height when the triangle starts at the observer’s eye.

Worked example

The 45° line rises 10 m from eye level. The extra 1.5 m below that line is part of the tower height.
  1. At a horizontal distance of 10 m and elevation angle 45°, the top is 10 m above eye level.
  2. With eyes 1.5 m above the ground, the tower is 11.5 m high.

Ask your child five quick questions.

One per step. The answer and the reason are given.

  1. Which reference direction do you measure from when looking towards the door’s top?

    The horizontal through your eyes. The vertical frame is not the starting direction for a sight angle.

  2. Which measurement completes the building height after you find the rise above your eyes?

    Your eye height above ground. The calculated rise leaves out the section below eye level.

  3. What must you remove from the combined height to find the flagstaff’s height?

    The building’s height. The requested part begins at the roof, but the combined height begins at ground.

  4. Which part of the longer shadow measures 2 m?

    The part beyond the shorter. Both shadows include the stretch nearest the tower.

  5. How do you combine the distances when the pole stands between the endpoints?

    Add the horizontal distances. The two bases cover the width without overlapping.

15 more questions with answers, 15 with pictures