Find the value of `sec^(-1)2+sin^(-1)(1/sqrt2)`

Options

A. `(5pi)/(12)`
B. `(7pi)/(12)`
C. `pi/4`
D. `(3pi)/4`

Answer

To find the value of \( \sec^{-1}(2) + \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \), we can follow these steps:

### Step 1: Evaluate \( \sec^{-1}(2) \)

The function \( \sec^{-1}(x) \) gives us the angle \( \theta \) such that \( \sec(\theta) = x \).

For \( \sec^{-1}(2) \):
\[ \sec(\theta) = 2 \implies \theta = \frac{\pi}{3} \]
This is because \( \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2 \).

### Step 2: Evaluate \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \)

The function \( \sin^{-1}(x) \) gives us the angle \( \phi \) such that \( \sin(\phi) = x \).

For \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \):
\[ \sin(\phi) = \frac{1}{\sqrt{2}} \implies \phi = \frac{\pi}{4} \]
This is because \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).

### Step 3: Add the two angles

Now we can add the two angles we found:
\[ \sec^{-1}(2) + \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{3} + \frac{\pi}{4} \]

### Step 4: Find a common denominator and simplify

To add \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \), we need a common denominator. The least common multiple of 3 and 4 is 12.

Convert each fraction:
\[ \frac{\pi}{3} = \frac{4\pi}{12} \]
\[ \frac{\pi}{4} = \frac{3\pi}{12} \]

Now add them:
\[ \frac{4\pi}{12} + \frac{3\pi}{12} = \frac{7\pi}{12} \]

### Final Answer

Thus, the value of \( \sec^{-1}(2) + \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) is:
\[ \frac{7\pi}{12} \]