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
Result and synced visuals
Unit circle
Projection lines show cosθ and sinθ. Special angles are marked for quick reference.
| 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.
How to use the unit circle explorer
Start with sine or cosine and leave A=1, b=1, φ=0, and D=0 to see the parent curve. Then change one transformation at a time so the unit-circle point, graph, derived period, and phase shift stay easy to explain.
What each control changes
- A changes amplitude for sine/cosine and vertical scale for tangent.
- b controls period; larger |b| repeats faster, while b=0 becomes a constant value.
- φ shifts the input angle, and the radians/degrees switch changes entry units without changing the formula meaning.
- D moves the midline up or down.
Classroom workflow
Use the θ slider or animation to connect cosθ and sinθ on the circle with the matching point on the graph. Copy LaTeX for notes, export CSV samples for spreadsheet checks, and turn on teacher mode when you want discussion prompts.
Common checks
- Tangent has vertical asymptotes where cosθ is zero, so the graph is discontinuous by design.
- Compare another graphing tool only after matching A, b, φ, D, angle units, and the x-domain.
- Use expressions such as pi/4 or sqrt(3)/2 when you want exact special-angle inputs.
See also
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.
How do amplitude, period, and phase shift change the graph?
A changes amplitude or tangent scale, b changes the period, φ shifts the angle input, and D moves the midline. Change one value at a time to connect the formula with the unit-circle point and graph.
How are radians and degrees handled?
You can enter φ and θ in radians or degrees. The explorer converts them consistently for the graph, unit-circle coordinates, derived period, and copied LaTeX.
Why does the tangent graph have breaks?
Tangent is sinθ/cosθ, so it is undefined where cosθ is zero. Those angles become vertical asymptotes, and the graph intentionally leaves breaks instead of drawing a false connecting line.