Graphs of the trigonometric functions

From the circle to a wave

Let the point $P$ go round the unit circle, and plot its height, $\sin x$, against the angle $x$. The height rises to $1$, falls to $-1$ and comes back, again and again. The result is a wave.

The graph of $y = \sin x$ shows:

• a wave that repeats every $2\pi$;
• zeros at $x = n\pi$;
• the highest value $1$ at $x = \dfrac{\pi}{2} + 2n\pi$, and the lowest value $-1$ at $x = \dfrac{3\pi}{2} + 2n\pi$;
• a half-turn symmetry about the origin, because sine is odd.

The graph of $y = \cos x$ is the same wave moved $\dfrac{\pi}{2}$ to the left. It is symmetric about the y-axis, because cosine is even.

The graph of tan x

• $\tan x = 0$ where $\sin x = 0$, at $x = n\pi$.
• As $x$ comes close to $\dfrac{\pi}{2}$ from below, $\cos x$ becomes small and positive while $\sin x$ is close to $1$. So $\tan x$ grows without end. The vertical line $x = \dfrac{\pi}{2}$ is an asymptote.
• The graph repeats every $\pi$, and it takes every real value.

The graphs of sec x, cosec x and cot x

$\sec x = \dfrac{1}{\cos x}$ equals $1$ where $\cos x$ equals $1$. It grows without end where $\cos x$ comes close to $0$. Its graph is a row of U-shaped branches, each at least $1$ above or $1$ below the x-axis. Cosec does the same with $\sin x$. The graph of $\cot x$ looks like that of $\tan x$, falling instead of rising, with asymptotes at the multiples of $\pi$.

Stretching and moving a wave

Many repeating motions follow $y = a\sin(bx + c) + d$, with $b > 0$.

• Amplitude $|a|$: half the distance from the lowest point to the highest point. A negative $a$ turns the wave upside down.
• Period $\dfrac{2\pi}{b}$: a larger $b$ squeezes the wave.
• Sideways shift: write $bx + c = b\left(x + \dfrac{c}{b}\right)$. The wave of $\sin bx$ moves by $-\dfrac{c}{b}$, to the right when this number is positive.
• Middle line $y = d$: the range is $[d - |a|, d + |a|]$.

For example, take $y = 3\sin\left(2x - \dfrac{\pi}{2}\right) + 1$. The amplitude is $3$ and the period is $\dfrac{2\pi}{2} = \pi$. The shift is $\dfrac{\pi/2}{2} = \dfrac{\pi}{4}$ to the right. The middle line is $y = 1$, so the range is $[-2, 4]$.

Set $b = 2$: two full waves now fit where one was. Then set $a = -2$ and watch the wave turn over.

A worked example: a giant wheel

A giant wheel has a radius of $10\ \text{m}$, and its centre is $12\ \text{m}$ above the ground. It turns once every $40\ \text{s}$. A seat is at the bottom at $t = 0$. Let us find the height $h$ of the seat at time $t$.

Step 1. Middle line and amplitude. The seat moves between $2\ \text{m}$ and $22\ \text{m}$. So the middle line is $12\ \text{m}$ and the amplitude is $10\ \text{m}$.

Step 2. Period. One turn takes $40\ \text{s}$, so $b = \dfrac{2\pi}{40} = \dfrac{\pi}{20}$ per second.

Step 3. The start. $-\cos$ starts at its lowest value, just like the seat. So $h = 12 - 10\cos \dfrac{\pi t}{20}$ metres.

Step 4. Check. At $t = 20\ \text{s}$, $\cos \pi = -1$, and $h = 12 + 10 = 22\ \text{m}$: the top, after half a turn.

Practise with new numbers

Try it in Python

This program draws $y = a\sin(bx + c) + d$ beside $y = \sin x$, and prints the amplitude, the period, the shift and the range.

Common mistakes

• The period of $\sin 2x$ as $4\pi$. It is $\dfrac{2\pi}{2} = \pi$. The wave is squeezed, not stretched.
• A negative amplitude. For $y = -5\sin x$ the amplitude is $5$. The minus sign turns the wave over.
• The wrong shift. In $\sin(2x - \pi)$ the shift is $\dfrac{\pi}{2}$, not $\pi$. Take $b$ out first: $\sin 2\left(x - \dfrac{\pi}{2}\right)$.
• Joining tan across an asymptote. The graph breaks at $x = \dfrac{\pi}{2}$, where there is no point at all.

Remember these

• $y = \sin x$ and $y = \cos x$ are waves between $-1$ and $1$, with period $2\pi$.
• $y = \tan x$ has period $\pi$ and asymptotes at the odd multiples of $\dfrac{\pi}{2}$.
• For $y = a\sin(bx + c) + d$: amplitude $|a|$, period $\dfrac{2\pi}{b}$, shift $-\dfrac{c}{b}$, middle line $y = d$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What are the amplitude and the period of $y = 4\sin 5x$?

2. Where does $y = \cos x$ cross the x-axis between $0$ and $2\pi$?

3. What is the greatest value of $y = 7 + 2\sin 3x$, and at what smallest positive $x$ is it reached?

4. Between $-\pi$ and $\pi$, where is $y = \tan x$ not defined?

5. How is the graph of $y = \sin\left(x - \dfrac{\pi}{3}\right)$ related to the graph of $y = \sin x$?