# Period of sine and cosine graphs and their equations

Connection between period of graph, equation and formula

#### What is the period of a sine cosine curve?

The Period is how long it takes for the curve to repeat.

As the picture below shows, you can 'start' the period anywhere, you just have to start somewhere on the curve and 'end' the next time that you see the curve at that height.

#### So, what is the formula for the period?

If you look at the prior 3 pictures, you might notice a pattern emerge.. The period has a relationship to the value before the $$\theta$$

This pattern is probably easiest to see if we make a table.

Equation Period Picture
$$y = sin ( \color{red}{1}\theta )$$ $$\color{red}{2} \pi$$
$$y = sin ( \color{red}{2}\theta )$$ $$\color{red}{ 1 }\pi$$
$$y = sin ( \color{red}{\frac{1}{2}}\theta )$$ $$\color{red}{ 4\pi }$$
$$y = sin ( \color{red}{4}\theta )$$ $$\color{red}{ \frac{1}{2} \pi }$$

Can you guess the general formula?

As you might have noticed there is a relationship between the coefficient in front of $$\theta$$ and the period. In the general formula, this coefficient is typically labelled as 'a'.

The general formula for $$sin( \color{red}{a} \theta )$$ or $$cos( \color{red}{a} \theta )$$ is

$period = \frac{2 \pi}{ \color{red}{a}}$

### Practice Problems

To solve these problems, just start at the x-axis and look for the first time that the graph returns to that 'height.' So, in this case, we're looking for the time when the graph returns to the -.5 value which is at $$2 \pi$$.

Remember: Find the height of the graph at the x-axis and then look for the first time that the graph returns to that height. In this case, the answer is $$\pi$$ or just $$\pi$$.

Graphs generated by http://www.meta-calculator.com

Remember: The formula for the period only cares about the coefficient, $$\color{red}{a}$$ in front of the x. The formula for the period is the coefficient is 1 as you can see by the 'hidden' 1:

$$-2sin( \color{red}{1}x)$$

$period = \frac{2 \pi}{ \color{red}{a}} \\ period = \frac{2 \pi}{1} \\ period = 2 \pi$

Remember: The formula for the period only cares about the coefficient, $$\color{red}{a}$$ in front of the x. The formula for the period is the coefficient is 8 :

$$-7 cos ( \color{red}{8}x)$$

$period = \frac{2 \pi}{ \color{red}{a}} \\ period = \frac{2 \pi}{ 8} \\ period = \frac{ \pi}{4}$

So, the big question here is: what do we do about the negative sign? Well, the answer is, we do not worry about the negative sign. Period tells us how long something is, and it must be a positive number.

$$3cos( -\color{red}{2}x)$$

$period = \frac{2 \pi}{ \color{red}{a}} \\ period = \frac{2 \pi}{2} \\ period = \pi$