Graphing in the Entire Coordinate Plane
The functions sine, cosine, and tangent are commonly graphed
in the coordinate plane, with x representing the
measure of an angle (the x units are usually given
in radians) and y measuring the value of a given
trigonometric function at that angle. The best way to see this is
to study the graphs themselves. In the image below, (a) is the graph
of y = sin x; (b) is y = cos x;
and (c) is y = tan x.
These graphs make evident a number of important characteristics
of trigonometric functions.
The domain of a function is simply the x values
for which the function can be calculated. In the case of the trigonometric
functions, the input is an angle measure, and the output is a ratio
hypotenuse, for example).
The domain of a trigonometric function can be seen in
its graph: it is the set of all x values for which
the function is defined. For sine and cosine, the domain is the
set of real numbers, because any angle measure has a sine and a
cosine; there are no values of x for which sine
or cosine doesn’t produce a y value.
The graph of the tangent function, however, tells a different
story. It shows certain x values, for which the
tangent is undefined. These undefined points occur when the cosine is
zero, since tan x = sin
and division by zero is undefined. The x-values
for which tangent is undefined show up on its graph as vertical
dotted lines every 180º, such that x = n(180º)
+ 90º, where n is an integer. For example, the
tangent function is undefined at the x value 2(180º)
+ 90º = 450º.
Like the domain, the range of the trigonometric functions
can be seen from their graphs. The range of a function is the set
of all possible values of the function. In other words, the range
is the set of all y values of the function.
The range of sine and cosine, as you can see in its graph
or by analyzing the unit circle, is –1 ≤ y ≤ 1.
The graphs of these two functions never rise above 1 or fall below
–1 and every point on the unit circle has a x and y value
between –1 and 1. Occasionally, you may see a question in which
the answer choices are possible values of sine or cosine. If any
of them are greater than 1 or less than –1, you can eliminate them.
The range of tangent is the set of real numbers. To see
why there are no bounds on the value of tangent, recall that the
) can get arbitrarily
close to zero, making the quotient get infinitely large.
The chart below summarizes what has been discussed in
the previous few paragraphs. We have also included the ranges and
domains of the other three trigonometric functions.
Sine, cosine, and tangent are all periodic functions,
meaning that their values repeat on a regular interval. This regular
interval is called the function’s period. Speaking more technically,
the period of a function is the smallest domain containing a full
cycle of the function. Take a look at the periods for sine, cosine,
- For y = sin x and y =
cos x, the period is 2π radians. This means that
every 360º, the values of sine and cosine repeat themselves. For
example, trigonometric functions of 0 and 2π radians produce the
- For y = tan x, the period
is π radians. Thus, the tangents of 0º and 180º are equal.
If a trigonometric function contains a coefficient in
front of x
, its period changes. In general, the
period of y
is the normal period of f
divided by b
For example, the period of y
= 8π radians.
Another useful property of the sine and cosine curves
(not tangent, though) is amplitude. The figure below shows the amplitude, a,
for the sine and cosine functions:
The amplitude of the sine and cosine functions is half
the vertical distance between its minimum value and its maximum
value. The amplitude of y = sin x and y =
cos x is 1 because the minimum and maximum values
of these functions are –1 and 1, respectively, and half the vertical
distance between these values is 1. The tangent graph has no amplitude,
because the tangent function has no minimum or maximum value. In
general, the amplitude of the trigonometric function y = af(x)
is |a|. The amplitude of 1
/3 cos x is
Here is an example of the type of problem you might see
concerning the graphs of the trigonometric functions.
is the period and amplitude of the function y =
As we just discussed, the period of y
is the normal period of f
divided by b
For the sine function, the normal period is 360o
In this example, b
= 3, so the period of this function
3 = 120o
In general, the amplitude of the sine function y
|. In this particular case, a
4. So, the amplitude is 4.
Here’s another example problem:
is the period and amplitude of the function graphed below?
To answer this question, you simply have to read the distances
off the graph. The function, which appears to be either a sine graph
or a cosine graph, repeats itself every 180o.
Its period is therefore 180o. Its minimum
and maximum values are ±1
/2, so its
amplitude is 1/
To handle any question about the graphs of trigonometric
functions, you should be able to answer questions about period and
amplitude based on the equation or graph of a given function.