`sin^(-1)((1)/(sqrt(2)))-3sin^(-1)((sqrt(3))/(2))=`

Options

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

Answer

To solve the expression \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \), we will follow these steps:

### Step 1: Calculate \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \)
The value of \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) corresponds to the angle whose sine is \( \frac{1}{\sqrt{2}} \).

We know that:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Thus,
\[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \]

### Step 2: Calculate \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \)
Next, we find \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \). This corresponds to the angle whose sine is \( \frac{\sqrt{3}}{2} \).

We know that:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]

### Step 3: Substitute the values into the expression
Now we substitute the values we found into the original expression:
\[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{4} - 3\left(\frac{\pi}{3}\right) \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ = \frac{\pi}{4} - \frac{3\pi}{3} \]
\[ = \frac{\pi}{4} - \pi \]
To combine these, we convert \( \pi \) to have a common denominator:
\[ = \frac{\pi}{4} - \frac{4\pi}{4} = \frac{\pi - 4\pi}{4} = \frac{-3\pi}{4} \]

### Final Answer
Thus, the final result is:
\[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) - 3\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = -\frac{3\pi}{4} \]