 # 7.02 Graphs of trigonometric functions

Lesson

## Graphs of trigonometric functions

The unit circle provides us with a visual understanding that the trigonometric functions of \sin \theta , \cos \theta and \tan \theta exist for angles larger than what can be contained in a right-angled triangle.

The unit circle is a circle of radius equal to 1 centred at the origin. By looking at the trigonometric ratios, we get the definitions of \sin \theta and \cos \theta on the unit circle as the x and y-values of a point on the unit circle after having been rotated by an angle of measure \theta in an anti-clockwise direction. Or, if \theta is negative, then the point is rotated in the clockwise direction.

Definition of \cos \theta and \sin \theta can extend beyond 0\degree \leq \theta \leq 90\degree .

The sine of the angle is defined to be the y-coordinate of the point on the unit circle.

The cosine of the angle is defined to be the x-coordinate of the point on the unit circle.

The tangent of the angle can be defined algebraically as the ratio \left(\dfrac{\sin \theta }{\cos \theta }\right). This also represents the gradient of the line that forms the angle \theta to the positive x-axis. It can also be geometrically defined to be the y-coordinate of a point Q, where Q is the intersection of the extension of the line OP and the tangent of the circle at \left(1,0\right).

As we move through different values of \theta the value of \cos \theta and \sin \theta move accordingly between -1 and 1.

### Exploration

The animation below shows this process for y=\sin \theta and y=\cos \theta as \theta travels around the unit circle.

The unit circle produces wave like curves for y=\sin \theta and y= \cos \theta .

If we plot the values of \sin \theta and \cos \theta according to different values of \theta on the unit circle, we get the following graphs:

y=\sin \theta:

y=\cos \theta:

The simplest way to calculate \tan \theta is to use the values in the above graphs to evaluate \dfrac{\sin \theta }{\cos \theta }, which gives us the following graph where the vertical lines are asymptotes:

Notice that all of these graphs are constructed with degrees on the horizontal axis. The function values behave in the same way as in the unit circle - for example, in the graph above of y=\cos \theta , we can see that it has negative y-values for all of the angles in the domain of 90\degree <\theta <180\degree . These are the values associated with the second quadrant where \cos \theta will be negative.

### Examples

#### Example 1

Consider the graph of y=\tan x given below.

a

Using the graph, what is the sign of \tan 340\degree?

A
Negative
B
Positive
Worked Solution
Create a strategy

Plot the point onto graph to determine the sign.

Apply the idea

The point \left(340\degree ,\tan 340\degree \right) is plotted on the graph.

Since the point is below thex-axis, the value of \tan 340\degree is negative.

b

Which quadrant does an angle with measure 340\degreelie in?

A
B
C
D
Worked Solution
Create a strategy

Consider what size angles are in each quadrant of the unit circle.

Apply the idea

On the unit circle, angles between 270\degree and 360\degree are in the fourth quadrant.

340\degree is between 270\degree and 360\degree which means it lies in the fourth quadrant, so the answer is option D.

Idea summary

The unit circle produces wave like curves for y=\sin \theta and y= \cos \theta .

\tan \theta = \dfrac{\sin \theta }{\cos \theta } which means the graph of y= \tan \theta will have asymptotes whenever \\ \cos \theta =0.

## Properties of trigonometric graphs

The graphs of y=\cos \theta and y=\sin \theta have certain common properties. Each graph demonstrates repetition. We call the graphs of y=\cos \theta and y=\sin \theta periodic or cyclic. We define the period as the length of one cycle. For both graphs, the period is 360\degree.

Below is an example of a cycle.

Because of the oscillating behaviour, both graphs have regions where the curve is increasing and decreasing. Remember that we say the graph of a particular curve is increasing if the y-values increase as the x-values increase. Similarly, we say the graph is decreasing if the y-values decrease as the x-values increase.

Below is an example of where y=\sin \theta is decreasing.

In addition, the height of each graph stays between y=-1 and y=1 for all values of \theta, since each coordinate of a point on the unit circle can be at most 1 unit from the origin. This means, the range of both the \sin\theta and \cos\theta functions is between -1 and 1.

### Examples

#### Example 2

Consider the curve y=\sin x drawn below and answer the following questions.

a

What is the x-value of the x-intercept in the region -360\degree <x<0^\degree ?

Worked Solution
Create a strategy

Use the fact that x-intercept is the point at which the curve intersects the x-axis.

Apply the idea

The x-intercept in this region -360\degree <x<0^\degree is plotted below.

The x-value at this point is x=-180\degree.

b

Determine whether the following statement is true or false.

As x approaches infinity, the graph of y=\sin x stays between y=-1 and y=1.

A
True
B
False
Worked Solution
Create a strategy

Use the fact that the graph of y=\sin x is periodic.

Apply the idea

The graph of y=\sin x is periodic, meaning that it will periodically return to the same y-values and will continue to oscillate between a height of -1 and 1.

So the answer is option A, true.

c

In which two of the following regions is the graph of y=\text{sin }x increasing?

A
90\degree<x<270\degree
B
-270\degree<x<-90\degree
C
270\degree<x<450\degree
D
-450\degree<x<-270\degree
Worked Solution
Create a strategy

Use the graph and choose the regions where the y-values increase from left to right.

Apply the idea

Each region of values is shown on the graph below:

The graph is increasing for the regions in options C and D.

Idea summary

The graphs of y=\cos \theta and y=\sin \theta periodic or cyclic. For both graphs, the period is 360\degree.

-1 \leq \sin \theta \leq 1 and -1 \leq \cos \theta \leq 1 for all values of \theta.

### Outcomes

#### VCMMG368 (10a)

Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies.