Sine and cosine on the unit circle

The unit circle has its center at the origin and a radius of 1. Pick any point on it. Its x coordinate is cos θ and its y coordinate is sin θ, where θ is the angle measured counter-clockwise from the positive x-axis. Drag the point to see how they change.

Angle θ
cos θ = x
sin θ = y

Why does this work?

Drop a vertical line from the point to the x-axis and you get a right triangle. Its hypotenuse is the radius, which has length 1. From right-triangle trigonometry:

Because the hypotenuse is 1, there is nothing to divide by: the sides are the cosine and sine. The circle also lets us go past 90°, where a right triangle alone no longer makes sense. The coordinates simply become negative.

The Pythagorean theorem on the same triangle gives the most important identity in trigonometry: cos²θ + sin²θ = 1. Check it in the readout above for any angle.

Special angles to remember

θ0°30°45°60°90°
cos θ1√3/2 ≈ 0.87√2/2 ≈ 0.711/20
sin θ01/2√2/2 ≈ 0.71√3/2 ≈ 0.871

Notice that the sine row is the cosine row read backwards: sin θ = cos(90° − θ).

Try this

  1. Set θ = 0°, 90°, 180° and 270°. Where is the point each time, and what are cos and sin?
  2. Find two angles with sin θ = 1/2. (Hint: the point must be at height 1/2.)
  3. Compare 30° and 150°. Which value stays the same and which changes sign? Why?
  4. In which quadrants is cos θ negative? In which is sin θ negative?
  5. Press Spin and watch the graph. Why are sine and cosine the same wave, just shifted by 90°?
Show answers
  1. 0°: (1, 0). 90°: (0, 1). 180°: (−1, 0). 270°: (0, −1). Cosine is the first number and sine is the second.
  2. 30° and 150°.
  3. Sine stays 1/2, cosine changes from +√3/2 to −√3/2. The two points are mirror images across the y-axis: same height, opposite x.
  4. cos θ < 0 in quadrants II and III (left half). sin θ < 0 in quadrants III and IV (bottom half).
  5. Cosine starts at the top (1) and sine starts at 0. Sine reaches the top a quarter turn (90°) later, so cos θ = sin(θ + 90°).