Download circular functions


How many radians are in 180°?
π or approximately 3

How many degrees in 2π radians?
360°

How many degrees in 1 radian?
57.2957…°

Find sin 47°
0.7314

Find sin 47.
0.1235


Learn about the circular functions and their
relationship to trigonometric functions.
Learn that their arguments are real numbers
without units rather than degrees.

Circular functions

Standard position


In many real-world situations, the independent
variable of a periodic function is time or distance,
with no angle evident. For instance, the normal daily
high temperature varies periodically with the day of
the year.
Circular functions are periodic functions whose
independent variable is a real number without any
units. These functions are identical to trigonometric
functions in every way except for their argument.
They are more appropriate for real-world
applications.

Two cycles of the graph of the parent cosine function
are completed in 720° or 4π units, because 4π units,
because 4π radians correspond to two revolutions.

To see how the independent variable can represent a real number,
imagine the x-axis from an xy-coordinate system lifted out and
placed vertically tangent to the unit circle in a uv-coordinate
system with its origin at the point (u, v) – (1, 0). Then wrap the xaxis around the unit circle. x = 1 maps onto an angle of 1 radian,
x = 2 maps onto 2 radians, x = π maps onto π radians, and so on.


The distance x on the x-axis is equal to the length on the unit circle.
This arc length is equal to the radian measure for the corresponding
angle. Thus the functions sin x and cos x for a number x on the xaxis are the same as the sine and cosine of an angle of x radians.
The graph show an arc of length x on the unit circle, with the
corresponding angle. The arc is in standard position on the unit
circle, with its initial point at (1,0) and its terminal point at (u, v).
The sine and cosine of x are defined in the same way as for the
trigonometric functions.
horiz.coordinate u
cosx 
 u
radius
1
vert.coordinate v
sin x 
 v
radius
1

The name circular function comes from the fact that x equals the
length of an arc on the unit circle. The other four circular functions
are defined as ratios of sine and cosine.
DEFINITION: Circular Functions
If (u, v) is the terminal point of an arc of length x in standard position
on the unit circle, then the circular functions of x are defined as
sin x = v
cos x = u
sin x
tan x 
cos x
cosx
cot x 
sin x
1
sec x 
cos x
1
csc x 
sin x


Circular functions are equivalent to trigonometric functions in
radians. This equivalency provides an opportunity to expand the
concept of trigonometric functions. Trigonometric functions were
first defined using the angles of a right triangle and later expanded
to include all angles.
The concept of trigonometric functions includes circular functions,
and the functions can have both degrees and radians as
arguments. The way the two kinds of trigonometric functions are
distinguished is by their arguments. If the argument is measured
in degrees, Greek letters represent them such as sin θ. If the
argument is measured in radians, the functions are represented by
letters from the Roman alphabet, for example, sin x)
Plot the graph of y = 4 cos 5x on your grapher, in radian mode.
Find the period graphically and algebraically. Compare your results.
Tracing the graph, you find that the first high point beyond x = 0 is
between x = 1.25 and x = 1.3. So graphically the period is between
1.25 and 1.3
To find the period algebraically, recall that
the 5 in the argument of the cosine function
is the reciprocal of the horizontal dilation.
The period of the parent cosine function is
2π, because there are 2π radians in a
complete revolution. Thus the period of the
given function is
1
(2 )  0.4  1.2566
5
The answer found graphically is close this
exact answer.
Find a particular equation for the sinusoid function graphed in the
figure below. Notice that the horizontal axis is labeled x, not θ,
indicating the angle is measured in radians. Confirm your answer by
plotting the equation on your grapher.
y = C + A cos B(x – D)
Write the general sinusoidal
equation using x instead of θ
Sinusoidal axis is y = 3,
so C = 3
Find A, B, C, and D using information from the
graph
Amplitude is 2, so A = 2
From one high point to the next is 11 – 1.
Period is 10.
10 5

B is the reciprocal of the horizontal dilation.
Dilation is
or , soB  
2 
5
Phase displacement is 1
Cosine starts a cycle at a high point.
(for y = cos x, so d = 1

y  3  2 cos (x  1) Write the particular equation.
5
Plotting this equation in radian
mode confirms that it is correct.

Sketch the graph of y = tan x
6
SOLUTION: In order to graph the function, you need to identify its
period, the location of its inflection points, and its asymptotes.
Period =
6
  6

Horizontal dilation is the reciprocal of
period of the tangent is π
6

; the
For this function, the points of inflection are also the x-intercepts, or
the points where the value of the function equals zero. So

x  0,  , 2 ...
6
x  0,6,12...
Asymptotes are at values where the function is undefined. So

  3 5
x   , , , ,...
6
2 2 2 2
x  3,3,9,15...
Recall that halfway between a point of inflection and an asymptote
the tangent equals 1 or -1. The graph illustrates these features.
Note that in the graphs of circular functions the number π appears
in either in the equation as a coefficient of x in the graph as a scale
mark on the x-axis.