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 this calculator effectively
This guide helps you use Unit Circle & Trigonometry Explorer with worked steps in a repeatable way: define a baseline, change one variable at a time, and explain each output using explicit assumptions before sharing results.
How it works
The calculator applies deterministic formulas to your input values and only rounds at the final display layer. This makes it useful for comparative analysis: keep one scenario as a baseline, then vary assumptions and measure the delta in both absolute terms and percentage terms. If a change appears too large or too small, verify units, period conventions, and sign direction before interpreting the result.
When to use
Use this page when you need a fast planning estimate, a classroom check, or a reproducible scenario that teammates can review. It is most effective at the decision-prep stage, where you need to compare options quickly and decide which assumptions deserve deeper modeling or external validation.
Common mistakes to avoid
- Mixing units such as percent vs decimal, or monthly vs yearly settings.
- Changing multiple fields at once, which hides the real cause of result movement.
- Comparing outputs across tools without aligning constants and default conventions.
- Treating rounded display values as exact inputs for downstream calculations.
Interpretation and worked example
Start with a baseline case and save that output. Next, edit one assumption to reflect your realistic alternative, then compare both the direction and size of change. If the direction matches domain intuition and magnitude is plausible, your setup is likely coherent. If not, check hidden defaults, unit conversions, boundary conditions, and date logic before drawing conclusions.
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.
What should I enter first?
Start with the minimum required inputs shown above the calculate button, then keep optional settings at their defaults for a first pass. After getting a baseline, change one parameter at a time so you can explain which assumption moved the output.
How precise are the results?
The calculator keeps internal precision and rounds only for display. Small differences can still appear if another tool uses different constants, period conventions, or rounding rules. Align assumptions before comparing final values.
Why can my result differ from another calculator?
Many tools choose different defaults for units, rate basis, date-count logic, and sign conventions. Verify those defaults first. If differences remain, use the worked example and compare each intermediate step to locate the branch that diverges.
How it’s calculated