Lesson

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 **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.

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.

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

b

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

A

First quadrant

B

Second quadrant

C

Third quadrant

D

Fourth quadrant

Worked Solution

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.

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.

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

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

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

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.

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