Pythagorean Identity Calculator
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Understanding Pythagorean Identities
What are Pythagorean Identities?
Pythagorean identities are fundamental relationships between trigonometric functions, derived from the Pythagorean theorem:
sin²θ + cos²θ = 1 (Primary Identity)
1 + tan²θ = sec²θ (Tangent Identity)
1 + cot²θ = csc²θ (Cotangent Identity)
Derivation from Unit Circle
These identities can be derived from the unit circle where:
- x = cos(θ) represents the x-coordinate
- y = sin(θ) represents the y-coordinate
- The radius = 1 (unit circle)
- x² + y² = 1 (circle equation)
Important Properties
Universal Truth
Valid for all angles where defined
Reciprocal Relations
sec θ = 1/cos θ, csc θ = 1/sin θ
Domain Restrictions
Consider function undefined points
Quadrant Independence
Works in all quadrants
Special Angles
| Angle | sin²θ | cos²θ | Sum |
|---|---|---|---|
| 0° | 0 | 1 | 1 |
| 30° | 1/4 | 3/4 | 1 |
| 45° | 1/2 | 1/2 | 1 |
| 60° | 3/4 | 1/4 | 1 |
| 90° | 1 | 0 | 1 |
Applications and Extensions
Double Angle Formulas
Used to derive sin(2θ) and cos(2θ)
Half Angle Formulas
Helps find sin(θ/2) and cos(θ/2)
Verifying Identities
Foundation for proving complex identities
Common Variations
- sin²θ = 1 - cos²θ
- cos²θ = 1 - sin²θ
- tan²θ = sec²θ - 1
- cot²θ = csc²θ - 1
Real-World Applications
Physics
Essential in wave mechanics and oscillatory motion
Engineering
Used in signal processing and structural analysis
Mathematics
Fundamental in calculus and advanced trigonometry