Chapter 5: Trigonometry

The chapter problem: the beacon and the rising tide

A coastguard station $S$ and a lighthouse $L$ stand $10$ km apart on a straight coast. A distress beacon $B$ is at sea. From $S$ , the angle between the line to $L$ and the line to $B$ is $30^\circ$ . From $L$ , the angle between the line to $S$ and the line to $B$ is $105^\circ$ . A rescue boat waits at $S$ , and the crew must know how far the beacon is from each station.

The harbour beside $S$ is a sector of a circle. Its two straight walls are $4$ km long and meet at $S$ at an angle of $0.9$ radians. The harbourmaster needs the length of the curved sea wall and the area of water the harbour holds.

The sea level at the harbour rises and falls with the tide. After $t$ hours the height $h$ metres of the water above mean sea level is modelled by

h = 12\cos\theta + 5\sin\theta,

where $\theta$ is measured in radians and turns through a full circle once every $12$ hours. The harbour gate closes when the water reaches $10$ m. The crew want to know the greatest height and the times when the gate closes.

By the end of the chapter you will answer all of this exactly. You will find both distances from the survey, and the area of the triangle. You will find the length and the area of the harbour sector. You will rewrite the tide model as one wave, find its greatest height, and solve for the times when $h = 10$ .

First contact — no working needed yet. The beacon is closer to $L$ than to $S$ , and the largest angle is at $L$ . Look at the triangle $SLB$ on rough paper. Which side is longest, and which is shortest? Estimate the distance $SB$ . Then read the tide model: guess its greatest value before you do any algebra. Write both estimates down.

If you are not sure where to start

The side opposite the largest angle is always the longest. The angle at $L$ is $105^\circ$ , so the side opposite it, which is $SB$ , is the longest side. The angle at $B$ is $180^\circ - 30^\circ - 105^\circ = 45^\circ$ . The side opposite $30^\circ$ is $LB$ , so $LB$ is the shortest.

For the tide, the two numbers $12$ and $5$ add to at most $12 + 5 = 17$ only when both are at their peak, and they do not peak together. A reasonable guess is between $12$ and $13$ . Section 5.9 turns the guess into an exact value.

By the end of this chapter you will be able to

use the sine rule and the cosine rule to find sides and angles in any triangle;
find the area of a triangle from two sides and the included angle;
measure an angle in radians, and convert between radians and degrees;
find the length of an arc and the area of a sector;
sketch and read the graphs of sine, cosine and tangent, and describe their periodicity;
state and use the exact values of sine, cosine and tangent at the special angles;
use small-angle approximations for sine, cosine and tangent;
work with secant, cosecant and cotangent, and with the inverse trigonometric functions;
use the trigonometric identities, including the addition and double-angle formulae;
write $a\cos\theta + b\sin\theta$ as a single sine or cosine;
solve trigonometric equations in a given interval without losing solutions;
prove a trigonometric identity;
use trigonometry with vectors, kinematics and forces.

Where it is used

Trigonometry is the mathematics of anything that repeats, turns or leans. Waves, tides, alternating current and sound are all sine curves. Forces and velocities are resolved into components with sine and cosine, which is how mechanics adds them. Radians are the natural unit for angles in calculus: the derivative of $\sin x$ is $\cos x$ only when $x$ is in radians. The identities in this chapter turn complicated expressions into simple ones, and the single-wave form is how engineers describe any oscillation that is a sum of two.

Your turn to connect it

Think of something that repeats or turns in your own life. It could be a swing, a turning wheel, a tide, a fan, or the length of daylight through the year. Write down the repeating quantity, the largest and smallest values you would expect, and one question about it that a graph could answer.

Chapter 5: Trigonometry

The chapter problem: the beacon and the rising tide

By the end of this chapter you will be able to

Where it is used

Your turn to connect it

Roadmap

Settings