sin²θ + cos²θ = 1 on a unit circle
Turn a radius of length 1 round a circle. Its height is sin θ and its distance across is cos θ. Change the angle, read sin, cos and tan, and see that sin²θ + cos²θ is always 1.
Core concept
On a circle of radius 1, a point at angle θ is cos θ across and sin θ up, so by Pythagoras sin²θ + cos²θ = 1 for every angle; tan θ = sin θ ÷ cos θ, which has no value at 90°
Topic: Trigonometry
Draw a circle of radius 1 with its centre at O. Turn a radius through an angle: The amount of turn between two lines that meet, measured in degrees (°). All key words θ from the line going right. The point at its end is cos θ across and sin θ up. The radius, the height and the distance across make a right-angled triangle with a hypotenuse of 1, so by Pythagoras sin²θ + cos²θ = 1², whatever the angle. The ratio: A way to compare two amounts, like 2 cups of rice to 3 cups of water (2 : 3). All key words tan θ = sin θ ÷ cos θ is the slope of the radius. At 90° the radius points straight up, cos θ = 0, and tan θ has no value. Past 90°, the point moves to the left, so cos θ becomes negative.
What you will learn
- Read sin θ and cos θ as the height and distance across on a unit circle.
- Show that sin²θ + cos²θ = 1 for different angles.
- Find tan θ as sin θ ÷ cos θ, and say why it has no value at 90°.
Before you start
- Know Pythagoras' theorem and the ratios sin, cos and tan in a right-angled triangle.
Materials
- A compass, a protractor and graph paper
Equipment
- A calculator
Procedure
- Record the angle 30°.
- Record 45° and 60°.
- Record 90°.
- Record 135°.
How this simulation is simplified
- The circle has radius 1; angles are measured anticlockwise from the line going right.