The Unit Circle

Notes/Exercise: Grade 11 Trigonometry Unit circle

When solving right angled triangles with SOH CAH TOA, we considered values of sine, cosine and tangent between zero and 90 degrees only, as follows:

GeoGebra link

However, values of sine, cosine and tangent are defined for angles between negative infinity and positive infinity.

The unit circle shown on the applet below allows us to explore trig values between zero and 360 degrees.

GeoGebra link

Notice that some trig values are positive and some are negative. We can now define the values of cosine and sine to be the (x,y) values of a point on the circumference of the unit circle.

Pause to play an estimation game:

Teacher vs Students Team A vs Team B Solo
Estimate sin,cos,tan to 360 T vs S Estimate sin,cos,tan to 360 A vs B Estimate sin,cos,tan to 360

In words:

Let P be a point on the circumference of a circle with radius one unit and center at the origin. Draw a radius from the center to the point P. Let \theta be the angle between the positive x axis and the radius, measured counter-clockwise. The coordinates of P are (\cos\theta,\sin\theta).

Remembering that \tan\theta=\frac{\sin\theta}{\cos\theta} gives us all three ratios. This leads to a simpler diagram for the values of sine and cosine:

GeoGebra link

Positive and Negative values of Sine, Cosine, Tangent

Values of cosine are defined as the x coordinates of the point P. These are positive on the right hand side of the diagram.

Values of sine are defined as the y coordinate of the point P. y coordinates are positive in the upper half of the diagram.

Values of tangent are defined as \frac{\sin\theta}{\cos\theta}=\frac{y}{x}. These are positive on one diagonal and negative on the other.

We refer to the quadrants of the Cartesian Plane as follows:

The following diagram is a helpful for remembering which trig values are positive where.

A simpler version of the diagram is often referred to as the CAST diagram:

 

Reference Angle

A reference angle is an angle in the first quadrant. In the following applet a point in the first quadrant is reflected to make a total of four points. With information from one of these points, we can calculate the sine, cosine and tangent of the other three.

Calculate the angle for each point, and compare the x, y coordinates given on the diagram with \cos and \sin values given for the angle on your calculator.

GeoGebra link

Practice

Practice calculating angles in other quadrants with this applet.

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