Trig Identity Checker
Verify a trigonometric identity numerically by sampling angles and reporting max deviation.
About Trig Identity Checker
Rather than attempting a symbolic proof, this checker evaluates both sides of a trigonometric identity at 100 sample angles and reports the maximum numerical deviation. A deviation under 1×10⁻¹⁰ means the identity holds to floating-point precision.
Formulas
Pythagorean Identity
sin²θ + cos²θ = 1
- θ
- Any angle
Double Angle (sine)
sin(2θ) = 2·sin θ·cos θ
- θ
- Any angle
Worked example
With the default values below, Trig Identity Checker returns Verdict of VERIFIED — identity holds to machine precision. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Identity to verifyone of sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ, sin(2θ) = 2·sin θ·cos θ, cos(2θ) = cos²θ − sin²θ, sin(−θ) = −sin θ, cos(−θ) = cos θ, tan θ = sin θ / cos θ
- sin²θ + cos²θ = 1
Results
- Verdict
- VERIFIED — identity holds to machine precision
- Max numerical deviation
- 2.2204 × 10⁻¹⁶
- Worst-case angle
- 162.3800 °
- Identity
- sin²θ + cos²θ = 1
Step by step
- 1Identity tested= sin²θ + cos²θ = 1
- 2Angles sampled= 100 angles across 0°–353°
- 3Max deviation= 2.2204e-16
- 4Verdict= Identity holds (deviation < 1e-10)
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Is numerical verification a real proof?
No — it is strong evidence but not a formal proof. However, for standard identities the deviation should be at floating-point error levels (< 10⁻¹⁴). A non-identity will show large deviation.
Why sample at irrational-step angles?
Using a non-round step avoids accidentally hitting only special angles where an identity might coincidentally hold but fail elsewhere.