Consider the following. x = sin 1/2 θ , y = cos 1/2 θ , -π ≤ θ ≤ π. Eliminate the parameter to find a Cartesian equation of the curve

Solution:

Given: x = sin(1/2)θ and y = cos(1/2)θ

⇒ x² = sin²(1/2)θ

⇒ y² = cos²(1/2)θ

So, x² + y² = sin²(1/2)θ + cos²(1/2)θ = 1.

x² + y² =1 ----->(1)

When θ = -π

⇒ x = sin (-π/2) = -1

⇒ y = cos (-π/2) = 0

When θ = π

⇒ x = sin (π/2) = 1

⇒ y = cos (π/2) = 0

We know that its a semi-circle between (-1,0) and (1,0).

But, there are 2 possibilities.

When θ = 0

⇒ x = sin 0 = 0

⇒ y = cos 0 = 1

So the semi-circle passes through (0,1).

In cartesian form, from(1) we have, x² + y² =1

⇒ y² = 1 - x²

⇒ y = √(1 - x²)

Therefore, the cartesian equation of the curve is y = √(1 - x²).

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Consider the following. x = sin 1/2 θ , y = cos 1/2 θ , −π ≤ θ ≤ π. Eliminate the parameter to find a Cartesian equation of the curve

Summary:

Consider the following. x = sin 1/2 θ , y = cos 1/2 θ , −π ≤ θ ≤ π. After eliminating the parameters, the cartesian equation of the curve is y = √(1 - x²).