Enter Values
Example Data Table
| Example | Input | Expected Relationship | Use Case |
|---|---|---|---|
| 30° angle | θ = 30 degrees | sin²θ + cos²θ = 1 | Basic angle verification |
| Known sine | sin(θ) = 0.6, Quadrant I | cos(θ) = 0.8 | Missing side relation |
| Known tangent | tan(θ) = 0.75, Quadrant III | 1 + tan²θ = sec²θ | Signed ratio reconstruction |
| Known secant | sec(θ) = 2, Quadrant IV | cos(θ) = 0.5 | Reciprocal identity check |
Formula Used
Main Pythagorean identity: sin²θ + cos²θ = 1
Tangent and secant identity: 1 + tan²θ = sec²θ
Cotangent and cosecant identity: 1 + cot²θ = csc²θ
Reciprocal formulas: cscθ = 1 / sinθ, secθ = 1 / cosθ, cotθ = 1 / tanθ
Residual: residual = left side − right side. A value near zero confirms the identity.
How to Use This Calculator
- Select angle mode when you know the angle.
- Select known ratio mode when you know one trig function value.
- Enter the angle, unit, ratio, quadrant, and precision.
- Press the calculate button.
- Review the identity table, residual values, and chart.
- Use CSV or PDF export for records, worksheets, or reports.
Pythagorean Trig Identities Guide
Why These Identities Matter
Pythagorean trig identities connect the main trigonometric functions through one simple idea. A point on the unit circle has coordinates cosθ and sinθ. Its distance from the origin is always one. That gives sin²θ + cos²θ = 1. From this base identity, the other two standard forms are created by division.
Identity Verification
This calculator checks whether both sides of each identity match. It also shows the residual. The residual is the difference between the left side and the right side. A residual close to zero means the identity is verified. Small decimal differences may appear because computers round long values.
Angle and Ratio Modes
Angle mode is useful when the angle is known. You may enter degrees or radians. The calculator finds sine, cosine, tangent, and their reciprocal functions. Ratio mode is helpful when one function value is given. Choose the quadrant so the correct signs are applied.
Quadrant Sign Control
Quadrants decide which values are positive or negative. In Quadrant I, all ratios are positive. In Quadrant II, sine and cosecant are positive. In Quadrant III, tangent and cotangent are positive. In Quadrant IV, cosine and secant are positive. Correct sign choice prevents false answers.
Advanced Learning Value
The result table makes each relationship visible. The graph compares the identity sides, so errors are easier to spot. Export options help teachers, students, and content creators save clean reports. This makes the tool useful for homework checks, lesson examples, worksheet creation, and exam review.
Best Practice
Use exact values when possible. Common examples include 0, 1/2, √2/2, √3/2, and 1. Decimal entries also work. Always check undefined cases. Tangent is undefined when cosine is zero. Cotangent is undefined when sine is zero. These cases are normal parts of trigonometry.
FAQs
1. What is a Pythagorean trig identity?
It is a trigonometric equation based on the unit circle. The main identity is sin²θ + cos²θ = 1. The other common identities are created by dividing this equation by cos²θ or sin²θ.
2. Why does the calculator show residual values?
The residual shows the difference between both sides of an identity. When the value is near zero, the identity is verified. Small nonzero values usually happen because of decimal rounding.
3. Can I use degrees and radians?
Yes. Choose degrees for common classroom angles. Choose radians for calculus, physics, and advanced math work. The calculator also displays converted angle values after calculation.
4. What happens if tangent is undefined?
Tangent is undefined when cosine equals zero. In that case, tangent and secant based checks may show undefined values. This is mathematically correct, not a calculator error.
5. Why is quadrant selection important?
Quadrants control signs. The same squared value can produce positive or negative ratios. Choosing the correct quadrant gives the correct sine, cosine, tangent, and reciprocal signs.
6. Can I calculate from a known trig ratio?
Yes. Select known ratio mode. Enter the function value and choose the quadrant. The calculator reconstructs related trig values using Pythagorean and reciprocal relationships.
7. What does secant mean?
Secant is the reciprocal of cosine. The formula is secθ = 1 / cosθ. It is undefined when cosine equals zero.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a formatted report that includes tables and a chart snapshot.