Graphing Sine & Cosine Functions

Sine and cosine graphs are repeating waves. Two numbers in $y = a\sin(bx)$ shape them: $a$ sets the amplitude (height) and $b$ sets the period (how quickly it repeats).

The period is $\dfrac{2\pi}{b}$ . A bigger $b$ squeezes more waves into the same space; the amplitude $|a|$ is the wave's height above and below the midline.

Amplitude and period

The amplitude is $|a|$ — how far the wave rises above and falls below its midline. The period is the horizontal length of one full cycle.

For $y = a\sin(bx)$ , the period is $\dfrac{2\pi}{b}$ . The plain sine wave has period $2\pi$ ; multiplying $x$ by $b$ divides that.

How b changes the wave

A larger $b$ shortens the period, packing the waves closer together. A $b$ between 0 and 1 stretches the wave out.

So to find the period, take $2\pi$ and divide by whatever multiplies the angle inside the function.

Worked examples

Example 1: finding the period

What is the period of $y = 4\sin 3\theta$ ?

Period is 2-pi over b

\dfrac{2\pi}{3}

Answer: $\dfrac{2\pi}{3}$

Example 2: the amplitude

What is the amplitude of $y = 4\sin 3\theta$ ?

Amplitude is the absolute value of a

|4|

Answer: $4$

Example 3: period when b is a fraction

What is the period of $y = 2\sin\left(\dfrac{1}{2}\theta\right)$ ?

Period is 2-pi over b

\dfrac{2\pi}{\tfrac{1}{2}}

Dividing by one half doubles it

4\pi

Answer: $4\pi$

Try one yourself

Common questions

How do I find the period?

Divide $2\pi$ by the coefficient $b$ of the angle: the period of $y = a\sin(bx)$ is $\dfrac{2\pi}{b}$ .

What is amplitude?

The height of the wave above and below its midline, equal to $|a|$ .

What does a larger b do?

It shortens the period, compressing more cycles into the same horizontal space.

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Graphing Sine & Cosine Functions

Amplitude and period

How b changes the wave

Worked examples

Example 1: finding the period

Example 2: the amplitude

Example 3: period when b is a fraction

Try one yourself

Common questions

Keep learning

Angles & Angle Measure

Translations of Trigonometric Graphs

Trigonometric Functions of General Angles