3.5 Circular Functions

Main Concepts:

Circular Functions: Periodic functions whose independent variable is a real number without units

We are accustomed to graphing period functions whose independent variable is theta, meaning theta ------> degrees

we will now graph periodic functions whose independent variable is a unitless real number, meaning x ------> radians

The numbers are unitless because a radian is a ratio in which the units cancel:

Radian=Radius/Circumference

Thus, using the common degrees to radian conversions (90=pi/2, 180=pi, 360=2pi etc.) we can label our independent variable axis (now x as opposed to theta) and find the period and phase displacement.

Note: The y-axis does not change because only the independent variable axis is measured in radians. Therefore, the period and phase displacement are notated differently, but the amplitude and sinusoidal axis remain the same.

Circular Function

Trigonometric function

(Blue=cos, Red=Sin)

As shown by these two graphs, circular functions are equivalent to trigonometric functions in radians. Thus, assuming that the terminal point of an arc on the unit circle is at (u,v) and the arc's length is x,

sin x=v cos x=u

tan x=sin x/cos x cot x=cos x/sinx

sec x=1/cos x csc x=1/sin x

Example Problem:

Given the equation y=3cos6(x-20)-2, find the sinusoid's amplitude, phase displacement, period, and sinusoidal axis.

1.First we must determine which of these sinusoidal characteristics is affected by the fact that the graph is in radians and not degrees.

Because the amplitude and sinusoidal axis are determined by the y-axis,

Amplitude=3

Sinusoidal axis: y=-2

2.We must then go about finding the phase displacement and the period.

In degrees, Period=360/6

Since the argument is measured in radians, Period=2pi/6

In degrees, Phase displacement=20

Since the argument is measured in radians, Phase displacement=20 radians

For more information:

http://colalg.math.csusb.edu/~devel/precalcdemo/circtrig/src/trgraphs.html

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