ScientistGrades 10–11⏱ 20 min
The unit circle and sine waves
Spin a point around a circle and watch its height draw the sine wave.
Standards: HSF-TF.A.2HSF-TF.B.5
1Wonder
Sound, light, ocean tides and the swing of a pendulum all rise and fall in the same smooth wave. Where does that wave come from?
Hint: it is hiding inside a circle.
2Explore
This activity needs a bigger screen
Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.
θ = 60° sin θ = 0.866 cos θ = 0.5
Units
Try this
- Drag the point to 90°. How tall is the red line? Now try 270°.
- Find two different angles that have the same sine.
- Press Spin and watch the wave draw itself. When does it start repeating?
- Switch to radians. What is a full turn in radians?
- Turn on the cosine wave. How is it related to the sine wave?
3Explain
Step 1 of 5
A circle of radius 1
The unit circle has its center at the origin and radius 1. Using radius 1 keeps all the numbers simple.
Why this matters
Because the right triangle inside the circle has hypotenuse 1, the Pythagorean theorem gives the most famous identity in trigonometry:
The Pythagorean identity
Useful values to remember:
| θ | 0 | 30° = π/6 | 45° = π/4 | 60° = π/3 | 90° = π/2 |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
4Practice
Question 1 of 5
What is sin 90°?
5Recap
- The unit circle has radius 1. The point at angle θ is (cos θ, sin θ).
- Radians measure angles with the radius: 180° = π, 360° = 2π.
- Plotting the height against θ draws the sine wave, which repeats every 2π.
- Cosine is the same wave shifted by π/2.
- for every angle.