Finding Relation between x, y and a

We are given the equations:

• $x = a\sin\theta$
• $y = a\cos\theta$

To find the relation, we first square both equations:

• $x^2 = (a\sin\theta)^2 \implies x^2 = a^2\sin^2\theta$
• $y^2 = (a\cos\theta)^2 \implies y^2 = a^2\cos^2\theta$

Next, we add the squared equations together:

$x^2 + y^2 = a^2\sin^2\theta + a^2\cos^2\theta$

Factor out $a^2$ from the right side:

$x^2 + y^2 = a^2(\sin^2\theta + \cos^2\theta)$

Using the fundamental trigonometric identity $\sin^2\theta + \cos^2\theta = 1$, we substitute this into the equation:

$x^2 + y^2 = a^2(1)$

This simplifies to the final relation:

$x^2 + y^2 = a^2$