Math Problem Statement

sin(0,5t)=x cos(0,5t)=y, make it a cartesian equation by eliminating the parameter

Solution

To eliminate the parameter tt from the parametric equations:

sin(0.5t)=xandcos(0.5t)=y,\sin(0.5t) = x \quad \text{and} \quad \cos(0.5t) = y,

we use the Pythagorean identity for sine and cosine:

sin2(θ)+cos2(θ)=1.\sin^2(\theta) + \cos^2(\theta) = 1.

Substituting sin(0.5t)=x\sin(0.5t) = x and cos(0.5t)=y\cos(0.5t) = y, we get:

x2+y2=1.x^2 + y^2 = 1.

Thus, the Cartesian equation is:

x2+y2=1.x^2 + y^2 = 1.

This represents a circle with a radius of 1, centered at the origin.


Would you like a more detailed explanation or examples? Here are five related questions to explore further:

  1. What if sin(kt)=x\sin(k t) = x and cos(kt)=y\cos(k t) = y with a different constant kk?
  2. How does the parameter tt influence the parametric equations' behavior over time?
  3. How can we extend this method to eliminate parameters in other trigonometric forms?
  4. Can you derive the Cartesian equation if the parametric equations involve phase shifts, such as sin(0.5t+ϕ)=x\sin(0.5t + \phi) = x?
  5. What does the equation x2+y2=1x^2 + y^2 = 1 represent geometrically in higher dimensions?

Tip: For trigonometric parametric equations, the Pythagorean identity is your key tool for eliminating parameters!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Parametric Equations
Pythagorean Identity

Formulas

sin^2(θ) + cos^2(θ) = 1

Theorems

Pythagorean Identity

Suitable Grade Level

Grades 9-12