Inverse Trigonometric Functions
Equations with Domain Limitations
Premium Question
Grade 12
Question:
The number of real solutions of $\tan^{-1}\sqrt{x(x + 1)} + \sin^{-1}\sqrt{x^2 + x + 1} = \frac{\pi}{2}$ is:
(1) $0$
(2) $1$
(3) $2$
(4) infinitely many
Step-by-Step Solution
Key Concept: For $\sin^{-1}\sqrt{x^2 + x + 1}$ to be defined, need $x^2+x \leq 0$, i.e. $x \in [-1, 0]$. For $\tan^{-1}\sqrt{x(x + 1)}$, need $x(x + 1) \geq 0$. Find the overlap and count.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)