Period Of A Cos Function

What is the flow of a sine cosine curve?

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

picture of period of sinx

As the movie beneath shows, you tin 'start' the menstruation anywhere, you just have to start somewhere on the curve and 'end' the side by side time that you come across the curve at that height.

picture of period of sinx diagram of graph of period for sin of half theta

Then, what is the formula for the catamenia?

If you look at the prior iii pictures, you might notice a design emerge.. The menstruum has a human relationship to the value before the $$ \theta $$.

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

Equation	Period	Motion-picture show
$$ y = sin ( \color{red}{1}\theta )$$	$$ \colour{red}{ii} \pi $$
$$ y = sin ( \colour{ruby-red}{ii}\theta )$$	$$ \color{carmine}{ ane }\pi $$
$$ y = sin ( \color{red}{\frac{one}{2}}\theta )$$	$$ \color{cerise}{ four\pi } $$
$$ y = sin ( \color{ruby-red}{4}\theta )$$	$$ \color{red}{ \frac{i}{2} \pi } $$

Tin you approximate the general formula?

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

The full general formula for $$ sin( \colour{red}{a} \theta )$$ or $$ cos( \color{ruby-red}{a} \theta )$$ is.

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

Practice Problems

Problem 1

Based on the graph below, what is the flow?

period picture 1

To solve these issues, simply outset at the 10-centrality and wait for the commencement time that the graph returns to that 'meridian.' So, in this example, nosotros're looking for the time when the graph returns to the -.v value which is at $$ 2 \pi$$.

Problem ii

Based on the graph below, what is the menses?

period picture 1

Recollect: Detect the meridian of the graph at the ten-axis and then look for the first fourth dimension that the graph returns to that superlative. In this instance, the answer is $$ \pi $$ or but $$ \pi $$.

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

Trouble 3

What is the period of the equation $$ y = -2sin(x) $$?

Recall: The formula for the period merely cares nigh the coefficient, $$ \color{red}{a} $$ in forepart of the ten. The formula for the period is the coefficient is 1 equally you can run across by the 'subconscious' one:

$ -2sin( \color{scarlet}{1}10) $

$ period = \frac{two \pi}{ \colour{cherry-red}{a}} \\ menstruation = \frac{ii \pi}{1} \\ menstruation = 2 \pi $

Trouble four

What is the period of the equation $$ y = -7cos(8x) $$?

Remember: The formula for the period only cares well-nigh the coefficient, $$ \color{scarlet}{a} $$ in front of the x. The formula for the period is the coefficient is viii:

$ -7 cos ( \color{cerise}{8}10) $

$ menstruation = \frac{ii \pi}{ \color{red}{a}} \\ catamenia = \frac{2 \pi}{ 8} \\ period = \frac{ \pi}{iv} $

Problem 5

What is the period of the equation $$ y = 3cos(-2x) $$?

And so, the big question here is: what do nosotros do nigh the negative sign? Well, the respond is, we do not worry almost the negative sign. Menstruation tells us how long something is, and it must be a positive number.

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

$ catamenia = \frac{ii \pi}{ \color{crimson}{a}} \\ period = \frac{2 \pi}{2} \\ period = \pi $

Demonstration
of Menstruum of sine Graph & Connection Unit Circle