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

Options

A. `(15pi)/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 break it down into two parts.

### Step 1: Find \( \sec^{-1}(2) \)
The secant function is defined as:
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \]
Thus, if \( \sec(\theta) = 2 \), then:
\[ \cos(\theta) = \frac{1}{2} \]
The angle \( \theta \) for which \( \cos(\theta) = \frac{1}{2} \) is:
\[ \theta = \frac{\pi}{3} \]
Therefore,
\[ \sec^{-1}(2) = \frac{\pi}{3} \]

### Step 2: Find \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \)
The sine function is defined as:
\[ \sin(\theta) = \frac{1}{\sqrt{2}} \]
The angle \( \theta \) for which \( \sin(\theta) = \frac{1}{\sqrt{2}} \) is:
\[ \theta = \frac{\pi}{4} \]
Thus,
\[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \]

### Step 3: Combine the results
Now, we can add the two results together:
\[ \sec^{-1}(2) + \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{3} + \frac{\pi}{4} \]

### Step 4: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12. Thus, we convert:
\[ \frac{\pi}{3} = \frac{4\pi}{12} \]
\[ \frac{\pi}{4} = \frac{3\pi}{12} \]

### Step 5: Add the fractions
Now we can add:
\[ \frac{4\pi}{12} + \frac{3\pi}{12} = \frac{7\pi}{12} \]

### Final Answer
The value of \( \sec^{-1}(2) + \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) is:
\[ \frac{7\pi}{12} \]

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