How does the explorer compute amplitude, period, and phase shift?

Amplitude is |A| for sine and cosine, while tan has no bounded amplitude. Period is 2π/|b| for sin and cos, and π/|b| for tan provided b ≠ 0. The phase shift is −φ/b when b ≠ 0, and the range becomes D ± |A| for sin/cos or all real numbers for tan.

What does the toggle for radians and degrees on φ control?

The φ input can be interpreted in radians or degrees. Switching units converts the typed value so that the current transformation stays in sync with the graph and the unit circle without recalculating from scratch.

Unit Circle & Trig Explorer — amplitude, period & phase

What you can explore

Control amplitude A, horizontal stretch b, phase φ (in radians or degrees), and vertical shift D in one form.
See derived quantities—amplitude, period, frequency, phase shift, and range—update instantly beside the main equation.
Synchronise the unit circle and function graph so θ, cosθ, sinθ, and y(θ) are always aligned.
Copy LaTeX, share the exact state through a URL, export CSV samples, or toggle an annotated teacher mode.

Set up the trig function

Function f(x)

Amplitude A

Horizontal stretch b

Phase φ

φ unit

Vertical shift D

Domain for x

θ (radians) Move the slider or animate θ to follow the point on the unit circle and graph.

θ = 0 rad (0°) Speed

Result and synced visuals

Unit circle

Projection lines show cosθ and sinθ. Special angles are marked for quick reference.

Derived quantities from the transformation
Amplitude (sin/cos)	—
Period T	—
Frequency 1/T	—
Phase shift C = −φ/b	—
Vertical shift D	—
Range	—

Graph of y(x)

How it’s calculated

Teacher notes

Connect amplitude and vertical shift directly to the peaks and midline before inviting students to move θ.
Pause animation at special angles to highlight cosθ and sinθ coordinates, then restart to show continuity.
Use the CSV export to plot the same curve in spreadsheets or graphing utilities for comparison exercises.

Frequently asked questions

How do I represent a constant function when b = 0?

Set b to 0 and choose any φ. The explorer evaluates y = A*f(φ)+D, so the graph collapses to a horizontal line and derived results show an infinite period and zero frequency.

Can I type expressions like π/4 or √3/2?

Yes. The fields understand common math expressions including pi, sqrt(), parentheses, and basic arithmetic, so π/4 or sqrt(3)/2 are parsed safely without using eval.

Unit Circle & Trigonometry Explorer with worked steps