Chapter 9 ยท Introduction

Chapter 9 โ€” Some Applications of Trigonometry

How tall is the school flagpole, the cell tower down the road, or the lighthouse across the bay? You do not need a ladder or a tape โ€” only an angle, a distance you can walk, and the trigonometric ratios of Chapter 8.

In 1802 the Great Trigonometrical Survey of India began to measure the subcontinent. Its surveyors could not climb the Himalayan peaks with a measuring tape. Instead they measured one long baseline on flat ground with great care, then looked at distant peaks through a theodolite and recorded angles. From a baseline and a few angles, trigonometry gave them every distance and every height โ€” including the heights of the highest mountains on Earth.

On a smaller scale you do the same thing when you stand some distance from a tower, tilt your head up and judge "about $45^\circ$". In Chapter 8 (trigonometric ratios) you learnt that in a right-angled triangle the ratios $\sin$, $\cos$ and $\tan$ depend only on the angle, and you met their exact values at $30^\circ$, $45^\circ$ and $60^\circ$. This chapter puts those ratios to work. Section 9.1 gives names to the lines and angles of an observation โ€” line of sight, angle of elevation and angle of depression โ€” and shows how to turn any scene into a right-angled triangle. Section 9.2 uses two triangles to find heights and distances when one distance is not known: two observation points, or a pole on a building.

Line of sight and horizontalโ†’Angle of elevation / depressionโ†’Right triangle: $\tan\theta=\dfrac hx$โ†’Two triangles, one unknownโ†’Heights and distances
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • define the line of sight, the angle of elevation and the angle of depression, and show that the depression of $B$ from $A$ equals the elevation of $A$ from $B$;
  • draw a clear figure for a word problem and mark the right angle, the given angle and the unknown;
  • choose between $\sin$, $\cos$ and $\tan$ and find a height or distance when the angle is $30^\circ$, $45^\circ$ or $60^\circ$;
  • solve problems with two observation points, on the same side or on opposite sides of a tower;
  • solve problems about a pole on a building, a rooftop observer, and two boats seen from a lighthouse;
  • give answers to a sensible precision, with $\sqrt3\approx1.732$ and $\sqrt2\approx1.414$ stated and units attached.
Teaser โ€” A Clinometer and a Tower

You stand $30$ m from the foot of a tower and read the angle of elevation of its top with a clinometer. Choose the reading with the buttons (or press Play to cycle through them) and watch the height of the tower that this reading implies: $h=30\times\tan\theta$. A bigger angle means a taller tower.

distance $x$ (m)โ€”
angle of elevation $\theta$โ€”
$\tan\theta$โ€”
tower height $h=30\tan\theta$ (m)โ€”

Sections in this chapter

9.1

Line of Sight, Angles of Elevation and Depression

The language of observation, why depression equals elevation, and how to turn a scene into a right-angled triangle and solve it.

$\tan\theta=\dfrac{h}{x}$, $\ \sin\theta=\dfrac hl$
Open section โ†’9.2

Heights and Distances

Two observation points, a pole on a building and two boats from a lighthouse: problems with two right triangles.

$h=\dfrac{d\tan\alpha\tan\beta}{\tan\beta-\tan\alpha}$
Open section โ†’
โœ… Before you start
You need the trigonometric ratios $\sin$, $\cos$, $\tan$ and their values at $30^\circ$, $45^\circ$, $60^\circ$ from Chapter 8: trigonometric ratios, the Pythagoras theorem (known from Class 9) to check right triangles, and equations of the form $\dfrac{h}{x}=\sqrt3$ and $\dfrac{x+20}{\sqrt3}=\sqrt3\,x$, which are linear equations (Section 3.2). From Class 9 you need alternate angles (equal when two lines are parallel) and the idea of a horizontal and a vertical line. Warm up: if $\tan\theta=\dfrac{12}{12\sqrt3}$, find $\theta$ (answer: $30^\circ$). Use $\sqrt3\approx1.732$ throughout.
โ† 8.3 Trigonometric Identities9.1 Line of Sight, Angles of Elevation and Depression โ†’