Trigonometry & Inverse Trigonometry
Inverse Trig Functions
nta_abhyas_2025
Grade 12
Question:
If $\sin^{-1}\frac{1}{4} + \sin^{-1}\frac{3}{5} = \sin^{-1}x$, then the value of $x$ is
$0$
$\frac{(3+\sqrt{7})}{20}$
$\frac{(3+\sqrt{7})}{20}$
$\frac{5}{6}$
Step-by-Step Solution
Key Concept: Use the addition formula for sine with inverse trigonometric functions
Given that $\tan^{-1}(1) + \sin^{-1}\left(\frac{2}{5}\right) = \sin^{-1}(x)$. Taking sine on both sides gives $\sin\left(\tan^{-1}(1) + \sin^{-1}\left(\frac{2}{5}\right)\right) = x$. This simplifies to $\left(\frac{1}{\sqrt{1-1}} + \frac{2}{5}\sqrt{1-\frac{1}{5}}\right) = x$.
Correct Answer: 3