Question

X = sin 1/2θ, y = cos 1/2θ, −π ≤ θ ≤ π(a) eliminate the parameter to find a cartesian equation of the curve.

Answer 1

First solve for theta in 'x' equation:
$\theta = 2 sin^{-1} x$

Substitute into 'y' equation:
$y = cos (\frac{2 sin^{-1} x}{2}) = cos (sin^{-1} x)$

Simplify by rewriting cos in terms of sine.
Using pythagorean identity: $sin^2 + cos^2 = 1$

$cos = \sqrt{1-sin^2} \\ \\ y = \sqrt{1-sin^2 (sin^{-1} x)}$
Simplify using inverse function property: $f(f^{-1} (x)) = x$

$y = \sqrt{1 - x^2}$

Answer 2

Answer:

$x^{2}+y^{2}=1$ Standard form.

Step-by-step explanation:

Parametric equations are useful to describe the motion of particles for instance, among other applications. In a trigonometric circle we can find its coordinates.

1) Rewriting for the sake of clarity:

$x=sin\frac{1}{2}\theta \:and\:\: y=cos\frac{1}{2}\theta \:,-\pi\leq \theta \leq \pi$

2) Solving, and using the Pythagorean Identity:

$x=sen\frac{1}{2}\theta ,y=cos\frac{1}{2}\theta \Rightarrow sen^{2}x+cos^{2}y=1\Rightarrow \left ( sen\frac{1}{2}\theta\right)^{2}+\left ( cos\frac{1}{2}\theta \right )^{2}=1\\\:Replacing\:the\:parameter\Rightarrow x^2+y^2=1$

Since the value for $\theta$ angle varies from $-\pi\leqslant\theta \leqslant \pi$ then this is a description of a half of a circumference.

Check the graph below.