Trigonometry Calculator – Quick Reference & Problem-Solving Framework
Evaluate trig functions, solve right triangles, and parse expressions fast without wading through a textbook. Use this framework to decide WHAT to compute, HOW, and how to sanity‑check the result.
1. Core Function Set
- sin θ = opp/hyp
- cos θ = adj/hyp
- tan θ = opp/adj
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
Inverse: asin, acos, atan return an angle (principal range). Use atan2(y,x) (not provided here) when quadrant matters.
2. Angle Modes & Conversion
- Radians = Degrees × π / 180
- Degrees = Radians × 180 / π
- Common: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2
3. Decision Steps
- Identify target: value of a function, missing side, or missing angle?
- Right triangle? If yes, pick ratio using given side/angle types (opp/adj/hyp).
- Need angle? Use inverse (asin/acos/atan) after computing ratio.
- Expression? Build it with sin(), cos(), etc. then evaluate.
- Validate: Magnitude in [-1,1] for sin/cos; tan/cot unbounded.
4. Shortcuts & Heuristics
- Use symmetry: sin(180°−θ)=sin θ, cos(180°−θ)=−cos θ.
- For small θ (radians): sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1−θ²/2.
- Right triangle solve: know 2 independent values → compute rest (Pythagorean + one trig ratio).
5. Guardrails
- Mode mismatch (degree vs radian) is #1 error – confirm before computing.
- Check domain: |input| ≤ 1 for asin/acos (after rounding tolerance).
- Round intermediate steps only at display time.
6. Common Pitfalls
- Using tan with adjacent & hyp (should be opp/adj).
- Forgetting that inverse trig returns principal angle (may need quadrant adjust).
- Feeding degrees directly into JavaScript Math.* (expects radians).
7. Quick Identity Reference
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
- tan θ = sin θ / cos θ
- sin(A±B)=sinA cosB ± cosA sinB
- cos(A±B)=cosA cosB ∓ sinA sinB
8. FAQ
Why am I getting NaN? Likely invalid expression or inverse domain breach >1 due to rounding.
When do I use atan vs atan2? atan2 handles full circle sign cases; this tool offers atan only—adjust using quadrant logic.
Best way to solve missing sides? Use the ratio involving only one unknown; else use Pythagorean first.
9. Action Tip
Before clicking calculate: write a 3‑symbol plan (e.g., “need a: use sin θ = opp/hyp”). That reduces errors and speeds checks.