Question

\(\sin^{-1}\frac{1}{\sqrt{5}} + \cos^{-1}\frac{3}{\sqrt{10}}\) is equal to

• A. \(\pi/6\)
• B. \(\pi/4\)
• C. \(\pi/3\)
• D. \(2\pi/3\)

Hint:

Convert inverse trig into angles and use identities.

Answer 1

The Correct Option is B

Concept: Let angles: \[ \sin A = \frac{1}{\sqrt{5}}, \quad \cos B = \frac{3}{\sqrt{10}} \]

Step 1: Find cos A.
\[ \cos A = \frac{2}{\sqrt{5}} \]

Step 2: Find sin B.
\[ \sin B = \frac{1}{\sqrt{10}} \]

Step 3: Use identity.
\[ \sin(A+B)=\sin A\cos B + \cos A\sin B \] \[ = \frac{1}{\sqrt{5}}\cdot\frac{3}{\sqrt{10}} + \frac{2}{\sqrt{5}}\cdot\frac{1}{\sqrt{10}} = \frac{5}{\sqrt{50}}=1 \] \[ \Rightarrow A+B=\frac{\pi}{2} \] Hence required value: \[ = \frac{\pi}{4} \]