X = sin 1 2 θ , y = cos 1 2 θ , −π ≤ θ ≤ π

Asked by maham237 @ in Mathematics viewed by 332 People

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

Answered by maham237 @


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}

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