Build sine and cosine graphs from rotation

Try

A rotating point begins at (1,0). Record its vertical coordinate after angles 0, π/2, π, 3π/2 and 2π. The values 0, 1, 0, −1 and 0 form five landmarks of one sine cycle. The horizontal coordinates 1, 0, −1, 0 and 1 form cosine landmarks.

Predict

Predict which graph starts at a maximum and which starts at a rising zero crossing. Then predict how y=3sin x+2 changes the five sine values. The outside multiplication scales heights; adding 2 shifts the reference line.

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.

Use the circle to confirm quarter-turn values. On paper, place angle along a new horizontal axis and plot the recorded sine heights. The new graph is not a circle: its horizontal coordinate is the input angle, not the point’s cosine coordinate.

For y=2cos(πt/2)+1, compute the following landmarks.

t	0	1	2	3	4
Output	3	1	−1	1	3

Join them with a smooth periodic curve. The midline is 1, amplitude 2 and period 4. Label units on t if it represents time.

Understand

For y=A sin(B(x−h))+k with nonzero A and B, amplitude is |A|, midline is y=k and period is 2π/|B|. The same quantities describe cosine. A negative A reflects the outputs about the midline; it does not give a negative amplitude. If A=0 or B=0, the function is constant and does not have the usual positive fundamental period of a sinusoid.

The phase shift must be found after factoring the inside coefficient. In sin(3x−π), write 3(x−π/3), so the horizontal shift is π/3, not π. A known original angle u occurs when x=h+u/B. This point mapping also works with negative B, which reverses input direction.

The range is [k−|A|,k+|A|] when the input covers the whole real line. A restricted input domain may attain only part of it. Periodicity means f(x+T)=f(x) for all allowed x in an unrestricted periodic model; seeing one repeated height alone does not identify a full period.

Apply

For each practice formula, mark the midline and five quarter-period points before sketching. Check a maximum, minimum and rising or falling crossing. These landmarks detect phase and reflection errors that a period calculation alone misses.

Continue

Ratios and reciprocals of circle coordinates introduce excluded inputs and vertical asymptotes. Their graphs need domain reasoning as well as periodic landmarks.