Trigonometric Graphs & Equations (Cambridge IGCSE Maths): Revision notes

Section 1

What do the graphs of sin x, cos x and tan x look like?

For $0° \leq x \leq 360°$ :

Function	Key features
$y=\sin x$	starts at 0, rises to 1 at 90°, falls to 0 at 180°, to -1 at 270°, back to 0 at 360°
$y=\cos x$	starts at 1, falls to 0 at 90°, to -1 at 180°, back to 0 at 270°, to 1 at 360°
$y=\tan x$	starts at 0, rises to infinity approaching 90° (undefined at 90°), repeats every 180°

Both $\sin x$ and $\cos x$ have a maximum of 1 and a minimum of -1, and repeat every 360° (their period). $\tan x$ is undefined at 90° and 270°, and has period 180°.

Key termsperiodamplitude

Exam tip

Sketch key points at 0°, 90°, 180°, 270°, 360° first, then join with a smooth curve — this avoids errors more than trying to plot the whole curve freehand.

Section 2

What symmetry do these graphs have?

$y=\sin x$ and $y=\cos x$ are symmetrical about their maximum and minimum points
$\sin(180° - x) = \sin x$ (this is the source of the sine rule's ambiguous case)
$\cos(360° - x) = \cos x$ and $\cos(-x) = \cos x$
$y=\tan x$ repeats identically every 180°: $\tan(x+180°) = \tan x$

Recognising these symmetries lets you find ALL solutions to a trig equation within a given range, not just the one your calculator gives first.

Section 3

How do I solve trigonometric equations?

To solve an equation like $\sin x = \frac{\sqrt{3}}{2}$ for $0° \leq x \leq 360°$ :

Find the principal value using the inverse function on your calculator
Use the symmetry of the graph to find any other solutions in the given range
Check both solutions lie within the stated range

For sine: if $\sin x = k$ gives principal value $x_1$ , the second solution is $180° - x_1$ . For cosine: if $\cos x = k$ gives principal value $x_1$ , the second solution is $360° - x_1$ . For tangent: solutions repeat every 180°, so the second solution is $x_1 + 180°$ (if still in range).

Key termsprincipal value

Example

$\sin x = \frac{\sqrt3}{2}$ : principal value $x_1 = 60°$ ; second solution $= 180° - 60° = 120°$ . Solutions: $x = 60°, 120°$ .

Section 4

How do I solve equations that need rearranging first?

Some equations must be rearranged into the form $\sin x = \ldots$ (or cos/tan) before solving.

Example: $2\cos x + 1 = 0 \Rightarrow \cos x = -\frac{1}{2}$ . Principal value from calculator: $x_1 = 120°$ . Second solution: $360° - 120° = 240°$ . Solutions: $x = 120°, 240°$ .

Always isolate the trig function on one side before applying the inverse function.

Common mistake

Forgetting the second solution entirely — always check the full 0°-360° range using the graph's symmetry, not just the calculator's single output.

Must Know

$\sin x$ and $\cos x$ have max 1, min -1, and repeat every 360°; $\tan x$ is undefined at 90°/270° and repeats every 180°
Sine second solution: $180° - x_1$ ; cosine second solution: $360° - x_1$ ; tangent second solution: $x_1 + 180°$
Always isolate the trig function before applying the inverse function
Always check both/all solutions lie within the given range
Sketch the key points at 0°, 90°, 180°, 270°, 360° to interpret graph shape questions
Never assume the calculator's answer is the only solution in the range

That's the notes covered.

Carry on to the next subtopic.