Mathematics 9709 · Pure Mathematics 1

Graphs of sin, cos and tan

You need these three graphs for everything else in this chapter.
Once you can draw them, the rest is easy. The period, the symmetry and the “second solution” of an equation all come from the shape. You do not memorise rules. You read them off the wave.

$\sin$ and $\cos$ are the same wave. Both have amplitude 1 and period $360^\circ$ . They just start at different points. $\tan$ is different. It has period $180^\circ$ , no amplitude, and an asymptote every $180^\circ$ .

The three shapes

$y=\sin x$ and $y=\cos x$ are smooth waves. They have amplitude 1 (they stay between $-1$ and $1$ ) and period $360^\circ$ .
Only the start is different. $\sin$ starts at $0$ . $\cos$ starts at its top, $1$ . So $\cos$ is $\sin$ moved left by $90^\circ$ .

y = sin x (solid) and y = cos x (dashed): same wave, cos shifted 90° earlier.

$y=\tan x$ is different. It has period $180^\circ$ (it repeats twice as often) and no amplitude. It has no top: it goes up with no limit.
The dashed lines at $90^\circ,\ 270^\circ,\ \dots$ are asymptotes. The curve gets closer and closer to them, but never touches them.

y = tan x: period 180°, with asymptotes (dashed) at 90° and 270°.

Stretches and shifts: y = a sin(bx) + c

In $y=a\sin(bx)+c$ each letter does one job:

$a$ sets the amplitude. $b$ sets the period $=\tfrac{360^\circ}{b}$ . $c$ sets the midline. You can read them straight off. You can also work them back from a printed graph.

$a$ is the amplitude. It makes the wave taller or shorter.
$b$ makes the wave thinner side to side. The period becomes $\tfrac{360^\circ}{b}$ , so a bigger $b$ means more waves.
$c$ moves the whole graph up. It is the midline, the line the wave goes up and down around.

Drag the sliders and watch each one work on its own. Make the wave taller with $a$ , add more waves with $b$ , and lift the whole graph with $c$ .

abc

y = 2 sin(2x) + 1·amplitude 2·period 180°·midline y = 1

The exam skill is doing this backwards. From a printed $y=a\sin(bx)+c$ , read the top and bottom for $a$ and $c$ , and count the waves for $b$ .

Worked example

From the graph, state $a$ , $b$ , $c$ for $y=a\sin(bx)+c$

This is 9709/12 Nov 2024 Q1. The printed curve has a top of $7$ and a bottom of $-1$ , and it fits two full waves into $0$ to $2\pi$ .

Amplitude is half the gap between top and bottom: $a=\tfrac{7-(-1)}{2}=4$ .
Midline is the average of top and bottom: $c=\tfrac{7+(-1)}{2}=3$ .
Two waves in a full turn means the period is halved, so $b=2$ .

Answer

a=4,\quad b=2,\quad c=3

Part (b): count the solutions without solving

The same question then asks how many solutions $4\sin(2x)+3=7-x$ has on $0\le x\le 2\pi$ . You do not solve it. You draw the straight line $y=7-x$ on top of your wave and count where they cross. The line starts high at $(0,7)$ and slopes slowly down to about $(2\pi,\ 0.7)$ . The wave has two humps and goes up to $7$ . The line crosses the wave five times. Here reading the graph is easier than algebra.

Where the marks go

Common mistake

Amplitude is half the distance from top to bottom, not the whole distance. Top

7

, bottom

-1

gives

a=4

, not

8

Common mistake

The period of

\tan x

180^\circ

, not

360^\circ

. Mixing up the two loses easy “state the period” marks.

Now you try

Write down the period and amplitude of $y=3\cos 2x$ . (This is the textbook's Worked Example 5.9 setup.) (9709-style)

Your turn— tap to reveal the worked answer (read off the stretch factors)

The $3$ in front is the amplitude. The $2$ inside cuts the period in half: $\tfrac{360^\circ}{2}=180^\circ$ .

Answer

\text{amplitude }3,\quad \text{period }180^\circ

PreviousExact values and angles of any size NextInverse trigonometric functions

Graphs of sin, cos and tan

You need these three graphs for everything else in this chapter.
Once you can draw them, the rest is easy. The period, the symmetry and the “second solution” of an equation all come from the shape. You do not memorise rules. You read them off the wave.

The three shapes

$y=\sin x$ and $y=\cos x$ are smooth waves. They have amplitude 1 (they stay between $-1$ and $1$ ) and period $360^\circ$ .
Only the start is different. $\sin$ starts at $0$ . $\cos$ starts at its top, $1$ . So $\cos$ is $\sin$ moved left by $90^\circ$ .

y = sin x (solid) and y = cos x (dashed): same wave, cos shifted 90° earlier.

$y=\tan x$ is different. It has period $180^\circ$ (it repeats twice as often) and no amplitude. It has no top: it goes up with no limit.
The dashed lines at $90^\circ,\ 270^\circ,\ \dots$ are asymptotes. The curve gets closer and closer to them, but never touches them.

y = tan x: period 180°, with asymptotes (dashed) at 90° and 270°.

Stretches and shifts: y = a sin(bx) + c

In $y=a\sin(bx)+c$ each letter does one job:

$a$ sets the amplitude. $b$ sets the period $=\tfrac{360^\circ}{b}$ . $c$ sets the midline. You can read them straight off. You can also work them back from a printed graph.

$a$ is the amplitude. It makes the wave taller or shorter.
$b$ makes the wave thinner side to side. The period becomes $\tfrac{360^\circ}{b}$ , so a bigger $b$ means more waves.
$c$ moves the whole graph up. It is the midline, the line the wave goes up and down around.

Drag the sliders and watch each one work on its own. Make the wave taller with $a$ , add more waves with $b$ , and lift the whole graph with $c$ .

abc

y = 2 sin(2x) + 1·amplitude 2·period 180°·midline y = 1

The exam skill is doing this backwards. From a printed $y=a\sin(bx)+c$ , read the top and bottom for $a$ and $c$ , and count the waves for $b$ .

Worked example

From the graph, state $a$ , $b$ , $c$ for $y=a\sin(bx)+c$

This is 9709/12 Nov 2024 Q1. The printed curve has a top of $7$ and a bottom of $-1$ , and it fits two full waves into $0$ to $2\pi$ .

Amplitude is half the gap between top and bottom: $a=\tfrac{7-(-1)}{2}=4$ .
Midline is the average of top and bottom: $c=\tfrac{7+(-1)}{2}=3$ .
Two waves in a full turn means the period is halved, so $b=2$ .

Answer

a=4,\quad b=2,\quad c=3

Part (b): count the solutions without solving

Where the marks go

Common mistake

Amplitude is half the distance from top to bottom, not the whole distance. Top

7

, bottom

-1

gives

a=4

, not

8

Common mistake

The period of

\tan x

180^\circ

, not

360^\circ

. Mixing up the two loses easy “state the period” marks.

Now you try

Write down the period and amplitude of $y=3\cos 2x$ . (This is the textbook's Worked Example 5.9 setup.) (9709-style)

Your turn— tap to reveal the worked answer (read off the stretch factors)

The $3$ in front is the amplitude. The $2$ inside cuts the period in half: $\tfrac{360^\circ}{2}=180^\circ$ .

Answer

\text{amplitude }3,\quad \text{period }180^\circ