Draw a point traveling around a circle of radius 1 and its height and shadow trace sine and cosine — even past 90°. The unit circle frees trigonometry from tiny triangles and lets it describe full rotations.
A circle of radius 1
Triangles only give angles between 0° and 90°. The unit circle sets trig free: put a circle of radius 1 at the origin, walk an angle theta counterclockwise from the positive x-axis, and the point you land on is exactly (cos theta, sin theta). Now sine and cosine make sense for any angle — 150°, 300°, even negative angles.
cos 45° = 0.71 · sin 45° = 0.71
The point on the circle is (cos θ, sin θ) — cosine runs along the ground, sine climbs the wall.
Sweep the angle around the whole circle. Watch cosine (the x-leg) and sine (the y-leg) grow, shrink, and flip sign as you cross the axes.
A radian measures the arc you walked: the full circle is 2π radians because the circumference of a radius-1 circle is 2π. So 180° = π and 90° = π/2. Calculus strongly prefers radians.
- Draw a unit circle and mark 0°, 30°, 45°, 60°, and 90°.
- Using the simulation (or the 45-45-90 and 30-60-90 triangle facts), fill in cos and sin at each mark.
- Now use symmetry: reflect your marks into the second quadrant (120°, 135°, 150°) and write those values with the correct signs.
- Check: sin²θ + cos²θ should equal 1 at every point. Verify it for 30° and 135°.
What you should see: You built the special-angle table from geometry and symmetry instead of memorizing it, and confirmed the Pythagorean identity.