Learn7357Paper 3 · Pure and StatisticsPure content (sections A–J)Graphs and transformations

Pure content (sections A–J) · Subtopic 14 of 66

Graphs of Sine, Cosine and Tangent

The shapes of $y=\sin x$ , $y=\cos x$ and $y=\tan x$ , their symmetries, and how $a$ , $b$ , $c$ and $d$ in $y=a\sin(b(x-c))+d$ stretch and shift them. Then use the graphs to count solutions and model things that repeat.

Cambridge 9709 Paper 1Edexcel YMA01 WMA11Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1AQA 7357 Paper 2≈ 45 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, with key points and asymptotes.
Find the amplitude, period, maximum and minimum of graphs like $y=a\sin bx+d$ .
Describe and sketch translations and stretches of trig graphs.
Use a graph to count the solutions of an equation, and to model periodic situations.

Pause and think

The point on the unit circle at angle $x$ is $(\cos x,\ \sin x)$ . As $x$ goes from $0^\circ$ to $360^\circ$ , describe what the height of the point does.

Section 2 of 14

Core idea

Start hereFerris wheels and tides

Ride a Ferris wheel and your height goes up, down, up, down, repeating every lap. The tide rises and falls twice a day. Both follow the same smooth wave: the sine curve. It's what you get when you unroll motion round a circle onto a straight time axis.

The big idea

$y=\sin x$ and $y=\cos x$ are waves of period $360^\circ$ ( $2\pi$ ) between $-1$ and $1$ ; $y=\tan x$ has period $180^\circ$ and vertical asymptotes. In $y=a\sin(bx)+c$ : $|a|$ is the amplitude, $\frac{360^\circ}b$ the period, $c$ the vertical shift.

1. From circle to wave

Unrolling the circle into the sine wave

x = 0

The solid curve is traced out as the angle grows. After

360^\circ

the point is back where it started, so the graph repeats.

As the point goes round the unit circle, its height is $\sin x$ . Plotting that height against the angle $x$ draws the sine wave. Its horizontal position, $\cos x$ , gives the cosine wave.

2. Key features

sin x, cos x and tan x

Cosine is sine shifted

90^\circ

left. Tangent has vertical asymptotes (dashed) at

90^\circ

and

270^\circ

, where

\cos x=0

and

\tan x=\tfrac{\sin x}{\cos x}

would divide by zero.

$\sin x$ : $0$ at $0^\circ$ , max $1$ at $90^\circ$ , $0$ at $180^\circ$ , min $-1$ at $270^\circ$ .
$\cos x$ : starts at $1$ ; it's $\sin x$ shifted $90^\circ$ left: $\cos x=\sin(x+90^\circ)$ .
$\tan x=\frac{\sin x}{\cos x}$ : undefined where $\cos x=0$ ( $90^\circ,270^\circ$ ): vertical asymptotes; period $180^\circ$ .

Key takeawaySymmetries you will use for equations

\sin(180^\circ-x)=\sin x

;

\cos(360^\circ-x)=\cos x

;

\tan(180^\circ+x)=\tan x

3. Transformations

y = a sin(bx): changing the amplitude

a = 0.5

Multiplying by

a

stretches the wave vertically: it now reaches

a

and

-a

. The zeros don't move.

y=a\sin(bx)+c

$|a|$: amplitude (height from middle line to peak).
$b$: squeezes horizontally: period $\frac{360^\circ}b$ or $\frac{2\pi}b$ .
$c$: shifts the middle line to $y=c$ .

Worked example $y=3\cos2x-1$ , $0\le x\le360^\circ$

State the amplitude, period, range, and the maximum and minimum points.

Goal: Key features for a sketch.
Strategy: Read $a$ , $b$ , $c$ ; locate max/min from where $\cos2x=\pm1$ .

$\text{amplitude }3,\quad\text{period }\tfrac{360^\circ}{2}=180^\circ$
$a=3$ , $b=2$ .
$-3-1\le y\le3-1\Rightarrow-4\le y\le2$
$\cos$ ranges from $-1$ to $1$ .
$\cos2x=1\text{ at }2x=0^\circ,360^\circ,720^\circ\Rightarrow x=0^\circ,180^\circ,360^\circ$
Max $y=2$ .
$\cos2x=-1\text{ at }2x=180^\circ,540^\circ\Rightarrow x=90^\circ,270^\circ$
Min $y=-4$ .

AnswerAmplitude

3

, period

180^\circ

, range

-4\le y\le2

; max

(0,2),(180^\circ,2),(360^\circ,2)

; min

(90^\circ,-4),(270^\circ,-4)

✓ Check

x=90^\circ

3\cos180^\circ-1=-4

✓.

Worked exampleFinding constants from a graph

$y=a+b\sin x$ has maximum $7$ and minimum $1$ , with $b>0$ . Find $a$ and $b$ .

Goal: $a$ , $b$ .
Strategy: Middle line is the average; amplitude is half the difference.

$a=\tfrac{7+1}2=4$
Middle line.
$b=\tfrac{7-1}2=3$
Amplitude.

Answer

a=4

b=3

✓ Check

4+3=7

4-3=1

✓.

Common misconception $\sin2x$ is not $2\sin x$

\sin2x

squeezes the wave horizontally (period halves);

2\sin x

stretches it vertically (height doubles).

Pause and think

How many solutions does $\sin2x=0.3$ have for $0\le x\le360^\circ$ ?

Summary

Know sin, cos, tan shapes and periods.
$a$ amplitude, period $360/b$ , $c$ shift.

Period: Length of one repeat.
Amplitude: Half the distance from min to max.
Asymptote: Line the graph approaches but never reaches.

HistoryWaves everywhere. In 1807 Joseph Fourier claimed that any repeating signal, however jagged, can be built by adding together sine waves. Mathematicians doubted him at first, but he was right. Today Fourier's idea runs inside every MP3, JPEG and phone call.

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