Trigonometric Identities
Verification Using Unit Circle Model
Explore how basic trigonometric identities can be verified using the unit circle model. Adjust the angle to see how the relationships between sine, cosine, and tangent change!
θ = 45°
sin²θ + cos²θ = 1
tanθ = sinθ/cosθ = 1
Observation:
For θ = 45°, the Pythagorean identity holds: sin²(45°) + cos²(45°) = 0.5 + 0.5 = 1. The ratio identity also holds: tan(45°) = sin(45°)/cos(45°) = 1.
The Mathematics Behind Trigonometric Identities
Key Concepts:
Trigonometric identities are equalities involving trigonometric functions:
- Pythagorean Identity: sin²θ + cos²θ = 1 (derived from unit circle equation x² + y² = 1)
- Ratio Identity: tanθ = sinθ/cosθ (definition of tangent)
- Reciprocal Identities: cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
Unit Circle Relationships:
In the unit circle (radius = 1), the x-coordinate is cosθ, the y-coordinate is sinθ, and the slope of the terminal side is tanθ. The tangent line intersects the x-axis at (1, 0) and forms a right triangle with the radius.
