# Trigonometry - Unit Circle

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Interactive and dynamic unit circle

Move the black point to change the angle and the blue point to change the hypotenuse of the enlarged triangle.

Turn parts of the representation on and off using the control panel to the left. >>

Click on points to reveal the hidden values.

Move the question mark to create your own questions.

## Pre-requisite 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 from the coordinates of the point on the circumference.

Download protractor on axis template.

## 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.

This is the only new knowledge that students will need.

This reduces cognitive load and frees up working memory to solve problems with trigonometry.

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

## Similar Triangles

Click and drag the blue point to enlarge the unit triangle. Click the points on the sides to reveal the missing lengths. Click and drag the question mark to set your own problems.

Use similar triangles to find the missing side. See the two methods below

## 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)

OR

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)

OR

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)

OR

## 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)

OR

## 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

OR

OR

## 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)

OR

## 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.

Compare the cognitive load needed to solve these problems using the unit circle compared to SOH CAH TOA.

## Unit Circle Tool

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

Click the points in the ratio 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.