# Periodic Functions

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Periodic functions are functions that repeat themselves. Below is the graph of the basic sine function.

Notice the flow of the graph from 0 radians to 2π(6.28). At 2π the graph begins to repeats itself and has

completed the duplication at 4π.

The graph will continue to duplicate every additional 2π or if measured in degrees every 360^{o}.

For instance the sin 3.6^{o} = .062790520 sin (3.6 + 360)^{o} = .062790520 sin (3.6 +720)^{o}= .062790520 All
= .062790520

This means the height on the vertical axis(above the horizontal axis) is always the same. In this case .062790520 and this causes the repetitiveness of the function.

Add 360^{o}(or 2π or a whole number multiple of either) to any angle and the value of the original trig function will not change.

For instance tan33.4^{o} = .6593785 The tan(33.4 + 360)^{o} also = .6593785 As will tan(33.4 + 5(360))^{o} However the period of the tan and cot functions is π

Use a calculator to confirm this results.

The below graph is of the sine function and the horizontal axis is marked off in radians. Starting with o at what number does the graph start to repeat itself? What is this number in degrees?
answer

Remember
- you can select any place on the graph, add 2π to the x value and the
graph will begin to duplicate itself at the new x value.
- For example the value of sin (π/6) = .5 radians. The value sin ((π/6)+ 2π) = .5 They are the same

meaning that the height above the x axis is the same and hence the graph begins to repeat.

If we used degrees instead of radians it would be the same. sin 30^{o} = .5 and the sin (30^{o} + 360^{o}) = .5

Hence we say that the basic sine function's period ( the point at which it begins to repeat) is 360^{o} or 2π

This next graph is of the cosine function and the horizontal axis is again marked off in radians. Starting with o radians at what number does the graph start to repeat itself? What is this number in degrees? (Try to answer these question before looking at the answers)
answer

Lets look at the tangent function
next

return PCTC