Parametric Trigonometric Equations

Trigonometry · Axiom Academy

LESSON Parametric Trigonometric Equations Using trigonometric substitutions to solve complex algebraic systems and revealing the hidden geometry of conic sections. 1. The Fundamental Example: The Unit Circle The most basic parametric trigonometric equations describe the unit circle: As the parameter t increases, the point (cos(t), sin(t)) traces out the unit circle. Notice how the algebraic equation emerges from the Pythagorean identity: By scaling the sine and cosine functions, we can create ellipses: The algebraic form emerges from the same Pythagorean identity: Therefore: (x/3)² + (y/2)² = 1 3. Using Trig Substitution to Solve Systems When faced with the constraint x² + y² = r², we can use the substitution x = r·cos(θ) and y = r·sin(θ) to simplify complex algebraic problems. Solution using parametric substitution: Let x = 5cos(θ) and y = 5sin(θ) (satisfies x² + y² = 25) Substitute into second equation: 5cos(θ) + 5sin(θ) = 5 Solve for θ to find the intersection points 4. Connection to All Conic Sections Different trigonometric substitutions correspond to different conic sections: sin²(t) + cos²(t) = 1 → circles and ellipses sec²(t) - tan²(t) = 1 → hyperbolas This connection between trigonometry and conic sections is fundamental to many areas of mathematics and physics, from planetary orbits to the paths of projectiles.

This is the written version of the interactive lesson above. See the full Trigonometry course.