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Period of sine and cosine graphs and their equations

Connection between period of graph, equation and unit Circle

What is the period of a sine cosine curve?

question
Answer
The Period is how long it takes for the curve to repeat.
picture of period of sinx
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.
picture of period of sinx

picture of period of sinx

diagram of graph of period for sin of half theta


diagram of graph of period for sin of half theta


diagram of graph of period for sin of half theta

So, how what is the formula for the period?

question
Answer
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
$$ y = sin ( \color{red}{1}\theta )$$ $$ \color{red}{2} \pi $$ picture of period of sinx
$$ y = sin ( \color{red}{2}\theta )$$ $$ \color{red}{ 1 }\pi $$ picture of period of sinx
$$ y = sin ( \color{red}{\frac{1}{2}}\theta )$$ $$ \color{red}{ 4\pi } $$ picture of period of sinx
$$ y = sin ( \color{red}{4}\theta )$$ $$ \color{red}{ \frac{1}{2} \pi } $$ picture of period of sine of 4x graph
Can you guess the general formula?
Answer
Practice Problems


problem 1) Based on the graph below, what is the period?
Answer
period picture 1

problem 2 Based on the graph below, what is the period?
Answer
period picture 1
Graphs generated by http://www.meta-calculator.com

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

problem 4 What is the period of the equation $$ y = -7cos(8x) $$ ?
Answer

problem 5 What is the period of the equation $$ y = 3cos(-2x) $$ ?
Answer

Demonstration of Period of sine Graph & Connection Unit Circle

Further Reading