Trigonometry

SOH-CAH-TOA in right-angled triangles, the exact values you must know, and the sine and cosine graphs.

7 min read

The ESAT trigonometry is right-angled-triangle based: SOH-CAH-TOA plus a small set of exact values. The sine and cosine rules are *not* required.

A right-angled triangle with angle theta, labelling the opposite, adjacent and hypotenuse sides. — Label the sides relative to the angle θ, then pick the ratio that uses the two sides you have.

$sin θ = \frac{opp}{hyp}, cos θ = \frac{adj}{hyp}, tan θ = \frac{opp}{adj} .$

Worked example

A right-angled triangle has hypotenuse $10$ and the angle $θ$ with opposite side $6$ . Find $θ$ .

$sin θ = \frac{6}{10} = 0.6 \Rightarrow θ = sin^{- 1} (0.6) \approx 37°.$

You must know these without a calculator:

$sin 0° = 0, sin 30° = \frac{1}{2}, sin 45° = \frac{1}{2}, sin 60° = \frac{3}{2}, sin 90° = 1.$
$cos$ is the same list reversed: $cos 0° = 1, cos 30° = \frac{3}{2}, \dots, cos 90° = 0.$
$tan 0° = 0, tan 30° = \frac{1}{3}, tan 45° = 1, tan 60° = 3 .$

Exam tip

Memory aid: write $sin$ of $0°, 30°, 45°, 60°, 90°$ as $\frac{0}{2}, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}$ . The tops are just $0$ to $4$ .

Graphs of y = sin x and y = cos x from 0 to 360 degrees, each oscillating between -1 and 1. — y = sin x and y = cos x repeat every 360° and stay between −1 and 1; cos is sin shifted left by 90°.

$sin x$ and $cos x$ oscillate between $- 1$ and $1$ with period $360°$ ; $tan x$ repeats every $180°$ and has asymptotes at $90°, 270°, \dots$ This symmetry is why an equation like $sin x = 0.5$ has more than one solution in a range.

Labelling opposite/adjacent before fixing which angle you are working from.
Forgetting a second solution because the trig graph is symmetric.
Reaching for the sine/cosine rule (not needed; build a right-angled triangle instead).

Trigonometry

Overview

Right-angled trigonometry

Exact values

The trig graphs

Common mistakes