Question

Find the equation of the curve \( (x, y) \) if \( \cos^{-1}(x-2) = \sin^{-1}(y+1) \).

Hint:

When dealing with inverse trigonometric functions, use the identity \( \cos^{-1} \theta + \sin^{-1} \theta = \frac{\pi}{2} \) to simplify the expression.

Answer 1

Step 1: Use the identity involving inverse trigonometric functions.
We are given the equation: \[ \cos^{-1}(x-2) = \sin^{-1}(y+1) \] We know that \( \cos^{-1} \theta + \sin^{-1} \theta = \frac{\pi}{2} \) for all \( \theta \) in the domain \( [0, 1] \). Using this identity, we can rewrite the equation as: \[ \cos^{-1}(x-2) + \cos^{-1}(y+1) = \frac{\pi}{2} \]
Step 2: Solve for \( x \) and \( y \).
Now, apply the identity \( \cos^{-1} \theta = \sin^{-1} \sqrt{1 - \theta^2} \) to express both terms in a solvable form, and we eventually get the equation of the curve: \[ (x - 2)^2 + (y + 1)^2 = 1 \] This represents a circle.