`cosec^(-1)((-2)/(sqrt3))=`

Options

A. `- pi/3`
B. `pi/3`
C. ` pi/2`
D. `- pi/2`

Answer

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

### Step 1: Understand the relationship between cosecant and cosine
The cosecant function is the reciprocal of the sine function. Therefore, we can express the cosecant inverse in terms of sine:
\[ \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \]
Thus, we have:
\[ \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) = \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \]

### Step 2: Find the angle whose sine is \(-\frac{\sqrt{3}}{2}\)
The sine function is negative in the third and fourth quadrants. The reference angle where \(\sin(\theta) = \frac{\sqrt{3}}{2}\) is \(\frac{\pi}{3}\). Therefore, the angles in the third and fourth quadrants are:
\[ \theta = \frac{4\pi}{3} \quad \text{(third quadrant)} \]
\[ \theta = -\frac{\pi}{3} \quad \text{(fourth quadrant)} \]

### Step 3: Choose the correct angle
Since we are looking for the principal value of \(\sin^{-1}\), we take the angle in the fourth quadrant:
\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]

### Step 4: Conclusion
Thus, we conclude that:
\[ \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) = -\frac{\pi}{3} \]

### Final Answer
\[ \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) = -\frac{\pi}{3} \]

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