Question

Write parametric equations for a circle of radius 2, centered at the origin that is traced out once in the clockwise direction for 0 ≤ t ≤ 4π. Use the module to verify your result. (Enter your answer as a comma-separated list of equations. Let x and y be in terms of t.)

Answer 1

Answer:

We know that equation of a circle with origin as it's center is given by

$x^{2}+y^{2}=r^{2}\\\\\therefore x^{2}+y^{2}=2^{2}\\\\(\frac{x}{2})^{2}+(\frac{y}{2})^{2}=1$

Comparing with $sin^{2}(\theta )+cos^{2}(\theta )=1$ we get

$\frac{x}{2}=sin(\theta )\\\\\therefore x=2sin(\theta )\\\\\frac{y}{2}=cos(\theta )\\\\\therefore y=cos(\theta )$

Since $sin(\theta ),cos(\theta )$ have a period of 2π but in the given question we need to increase the period to 4π  thus we reduce the argument by 2

$\therefore x=2sin(\frac{\theta }{2})\\\\y=2cos(\frac{\theta }{2})$