LearnYMA01WMA11 · Pure Mathematics 1TrigonometryGraphs of sine, cosine and tangent

Trigonometry · Subtopic 7 of 8

Graphs of Sine, Cosine and Tangent

Sine and cosine are waves with period $360^\circ$ ( $2\pi$ ) between $-1$ and $1$ ; tangent repeats every $180^\circ$ with asymptotes. Transform them like any graph: $a\sin x$ changes the height, $\sin bx$ squeezes the period to $\dfrac{360^\circ}b$ , and $\sin(x+c)$ slides it left.

Edexcel YMA01 WMA11≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Sketch $y=\sin x$ , $y=\cos x$ and $y=\tan x$ in degrees and radians.
Use exact values for $30^\circ$ , $45^\circ$ and $60^\circ$ .
Find the period, maximum and minimum of transformed graphs.
Use symmetry to count and find solutions.

Pause and think

A point goes round a circle of radius $1$ . Plot its height against the angle turned. What shape do you get, and when does it start repeating?

Section 2 of 14

Core idea

Start hereA Ferris wheel seat

Ride a Ferris wheel and plot your height against time: you get a wave. That wave is the sine graph, and because it repeats, an equation like $\sin x=0.6$ has many solutions, not one.

The big idea

$\sin$ , $\cos$ have period $360^\circ$ , $\tan$ period $180^\circ$ . Solve: find the principal value, use symmetry for the others in range. Identities: $\tan x=\frac{\sin x}{\cos x}$ , $\sin^2x+\cos^2x=1$ .

1. The graphs

The height of a point on a unit circle traces y = sin θ

θ (degrees) = 1

As the point goes round, its height rises to

1

90^\circ

, falls to

-1

270^\circ

and returns to

0

360^\circ

sin, cos and tan from 0° to 360°

Cosine is the sine wave shifted

90^\circ

left. Tangent shoots off to infinity at

90^\circ

and

270^\circ

The identities from the unit circle

$P=(\cos\theta,\ \sin\theta)$
Point on the unit circle at angle $\theta$ .
$\cos^2\theta+\sin^2\theta=1$
Pythagoras: distance from the origin is 1.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$
Gradient of the radius $OP$ .

2. Solving equations

Worked example $\sin x=0.6$ , $0\le x<360^\circ$

Solve.

Goal: All solutions.
Strategy: Principal value, then $180-x$ .

$x=\sin^{-1}0.6=36.9^\circ$
Calculator.
$x=180-36.9=143.1^\circ$
Symmetry of the sine curve.

Answer

36.9^\circ

143.1^\circ

✓ Check

\sin143.1^\circ=0.600

✓.

Worked example $\cos2x=-0.5$ , $0\le x<180^\circ$

Solve.

Goal: $x$ .
Strategy: Widen the range for $2x$ first.

$0\le2x<360^\circ$
Double the range.
$2x=120^\circ,\ 240^\circ$
$\cos^{-1}(-0.5)=120$ ; $360-120=240$ .
$x=60^\circ,\ 120^\circ$
Halve.

Answer

60^\circ

120^\circ

✓ Check

\cos240^\circ=-0.5

✓.

Worked example $2\sin^2x+\sin x-1=0$

Solve for $0\le x<360^\circ$ .

Goal: All $x$ .
Strategy: A quadratic in $\sin x$ .

$(2\sin x-1)(\sin x+1)=0$
Factorise like $2s^2+s-1$ .
$\sin x=\tfrac12:\ 30^\circ,150^\circ$
Two solutions.
$\sin x=-1:\ 270^\circ$
One solution (minimum point).

Answer

30^\circ

150^\circ

270^\circ

✓ Check

x=270

2-1-1=0

✓.

Common misconceptionDividing by sin x In

\sin x\cos x=\sin x

, factor out

\sin x

instead; dividing loses

\sin x=0

Pause and think

$\tan x=1$ , $0\le x<360^\circ$ ?

Summary

Periods 360°, 360°, 180°.
Principal value + symmetry.
Adjust ranges for $2x$ .

Period: Length after which a graph repeats.
Principal value: The calculator answer.

HistoryFrom chords to sines. Greek astronomers tabulated chords of circles. Indian mathematicians, notably Aryabhata around

500

AD, switched to half-chords, which are our sines. That change is why trigonometry looks the way it does today.

Fun factWaves everywhere. Every musical note is built from sine waves. Concert A vibrates

440

times a second, so its sound wave has period

\tfrac1{440}

of a second.

Keep reading for free

Create a free account to read the rest of this lesson: key formulae, worked examples, explore the graph, common mistakes, exam tip, guided practice and more. Your answers are checked and your progress is saved.

Create free account Sign in