Trigonometry - Unit Circle

Prerequisite knowledge

Pythagoras Theorem

Ratio Tables

Similar Triangles

The Unit Circle

The triangle has a constant hypotenuse of 1 unit.

The red and green shorter sides vary with the size of the angle.

Move the point on the circumference of the circle above to change the angle.

You can get the lengths of the shorter sides by reading off the values from the axis or click the point to reveal the coordinates.

Definitions of sine and cosine

Sine: sinθ is the length of the side opposite angle θ in a standard right angled triangle within the first quadrant of the unit circle

Cosine: cosθ is the length of the side adjacent to angle θ in a standard right angled triangle within the first quadrant of the unit circle. Cosine is also known as sine of the complement.

(cosθ,sinθ) is the coordinate of the point where the hypotenuse meets the circumference of the circle.

Students need to be able to see the unit triangle independent of the circle and in different orientations

Finding missing sides

Example 1 - Finding a missing shorter length given the hypotenuse is 3 and the angle is 30.

Draw the two similar triangles (above) and complete a ratio table (below)



alternatively ? = sin30 x 3/1

Example 2 - Finding the hypotenuse when given the shorter side is 3 and the angle is 30.

Draw the unit triangle to match the problem (left) and complete a ratio table (below)




alternatively ? = 1 x 3/sin30

Example 3 - Finding the shorter side when a given shorter side is 3 and the angle is 30.

Draw the unit triangle to match the problem (left) and complete a ratio table (below)



alternatively ? = sin30 x 3/cos30

Finding missing angles given the hypotenuse

Example 1 - Finding a missing angle given the hypotenuse is 3 and the opposite side is 2

Draw the unit triangle to match the problem (left) and complete a ratio table (below)




alternatively sin? = 2 x 1/3

Defining tangent

Tangent: tanθ is the height of the opposite side that is the tangent to the unit circle at the point x=1

tan θ = sinθ / cosθ

Finding missing angles without being given the hypotenuse

Example 1 - Finding a missing angle when given 2 shorter sides lengths using tanθ = sinθ / cosθ

Draw the unit triangle to match the problem (left) and complete a ratio table (below)

Example 2 - Finding a missing angle when given 2 shorter sides lengths using Pythagoras

Draw the unit triangle to match the problem (left).

Use Pythagoras to find the hypotenuse then complete a ratio table (below)




alternatively, use cosine

Exact trigonometric values

Finding exact values for 30 and 60 degrees

1) Draw an equilateral triangle with side lengths 1

2) Slice the triangle in half

3) Find the missing sides using Pythagoras

4) Compare with the unit triangle

Finding exact values for 45 degrees

1) Draw an isosceles right angled triangle shorter with side lengths 1

2) Find the missing side using Pythagoras

3) Compare with the unit triangle

4) Use the ratio table to find sin, cos and tan.


Unit Circle Tool

Click the image to access the full interactive tool on Desmos.

Click the points in the ration table to hide and reveal ratios.

Turn elements on and off using the control panel on the left. >>

Click the circle containing the folder icon to hide and reveal elements of the graph eg protractor and ratio table.