Equation of a Circle from parametric functions of sin and cos

Question

Given:

x = 2 cos (t/2) y = 2 sin (t/2)

How do we find the equation of the circle? I know that x^2 + y^2 = 1,

where x = cos(t) y = sin(t)

so x^2 = (2 cos (t/2))^2 y^2 = (2 sin (t/2))^2

How do you end up with x^2 + y^2 = 4?

Answer 1

HINT:

As $\displaystyle x=2\cos\frac t2,y=2\sin\frac t2,$

$\displaystyle \cos\frac t2=\frac x2, \sin\frac t2=\frac y2$

Use $\displaystyle \cos^2\frac t2+\sin^2\frac t2=1$ to elimiante $t$

Answer 2

We have always: $$\sin^2\alpha+\cos^2\alpha=1$$ so here we have then $$x^2+y^2=4$$

Answer 3

Generally speaking, it is $\cos^2{p} + \sin^2{p}=1$ for any real value of $p$. The value of $x^2+y^2$ is deduced from this, not the other way around.