Measure rotation and derive exact values

Try

A wheel of radius 3 cm turns through an arc of length 6 cm. Measure the turn by dividing arc length by radius: 6/3=2 radians. If a wheel twice as large makes the same turn, its arc is 12 cm and the ratio is still 2. Radians measure rotation independently of circle size.

Predict

Predict the arc-length ratio for a complete revolution. Circumference is 2πr, so the ratio is 2π. Thus 360 degrees equals 2π radians and 180 degrees equals π radians. Before changing the angle below, predict which coordinate becomes negative after passing a quarter-turn.

Experiment

Live experiment: one rotation, two coordinates

Before changing 30° to 150°, predict which coordinate changes sign. Then compare 30° with 390°: the point repeats, while the accumulated angle increases by a full turn.

Angle in degrees Rotate the radius Reset to 30°

0°30°90°150°180°270°360°390°

Angle: 30° = π/6 radians ≈ 0.5236 radians.

Coordinates: cos θ ≈ 0.866, sin θ ≈ 0.5.

Check: (0.866)² + (0.5)² ≈ 1. Displayed decimals are rounded; the identity holds exactly for a point on the unit circle.

This temporary experiment does not mark a lesson complete or save an assessment response. The curve is drawn from samples every 5°; the selected point is calculated at your exact input angle.

Inspect a quarter-turn, half-turn and negative quarter-turn. Record the horizontal and vertical coordinates, then compare an angle with that angle plus one revolution. The terminal point repeats, but the recorded rotation measure differs. Coordinates do not retain the number of complete revolutions.

Sketch a right triangle inside the first quadrant. For angle π/4, symmetry gives equal legs; on a unit circle their squares sum to 1. Both legs are therefore √2/2. For π/6 and π/3, bisect an equilateral triangle of side 2: the short leg is 1, the long leg √3, and the hypotenuse 2. Dividing by the hypotenuse gives exact unit-circle coordinates.

Understand

At angle θ, measured counterclockwise from the positive horizontal axis, the unit-circle point is (cos θ,sin θ). Cosine is the horizontal coordinate; sine is vertical. Clockwise rotation uses negative angles. Both coordinates lie between −1 and 1, and cos²θ+sin²θ=1.

At π/6 the point is (√3/2,1/2). At 5π/6 it is (−√3/2,1/2): reflection across the vertical axis changes only the horizontal sign. At 7π/6 both signs are negative. A reference angle provides magnitudes; the actual quadrant provides signs.

The formula s=rθ assumes θ in radians, matching length units for s and r, and positive radius. For signed rotation, rθ is a signed arc displacement; ordinary traveled arc length is nonnegative and uses the magnitude when motion follows one direction. Do not insert a degree number into this formula unchanged.

Apply

Derive the special-angle values before memorizing a table. In practice, enter only the requested rational coefficient when an angle is written kπ. Irrational coordinates use a choice or written argument rather than a decimal checker. Explain why coterminal angles need not be equal real numbers.

Continue

Next, unwrap the rotation into a graph whose horizontal axis records angle and whose vertical axis records one coordinate. This connects the circular picture to sine and cosine waves.