QuestionCalculate the value of $\csc^{-1}(-2)$ .

Studdy Solution

STEP 1

Assumptions1. The problem asks for the exact value of the inverse cosecant of -. . The range of the inverse cosecant function, $\csc^{-1}(x)$ , is $(-\infty, -\pi/] \cup [\pi/, \infty)$ .
3. The domain of the inverse cosecant function, $\csc^{-1}(x)$ , is $(-\infty, -1] \cup [1, \infty)$ .

STEP 2

We know that the cosecant function is the reciprocal of the sine function. So, we can rewrite the problem in terms of sine.
$\csc^{-1}(-2) = \sin^{-1}\left(\frac{-1}{2}\right)$

STEP 3

Now, we need to find the angle whose sine is $-1/2$ . We know that $\sin(\pi/6) =1/2$ , but since we want the negative value, we need to consider the angles in the third and fourth quadrants where the sine function is negative.

STEP 4

In the third quadrant, the angle that gives us a sine of $1/2$ is $\pi + \pi/6 =7\pi/6$ . In the fourth quadrant, the angle that gives us a sine of $1/2$ is $2\pi - \pi/6 =11\pi/6$ .

STEP 5

However, we need to remember the range of the inverse sine function, which is $[-\pi/2, \pi/2]$ . Neither $7\pi/$ nor $11\pi/$ fall within this range.

STEP 6

To get an angle in the range of the inverse sine function, we can subtract $2\pi$ from $\pi/6$ and $11\pi/6$ until we get an angle in the correct range.

STEP 7

Subtracting $2\pi$ from $7\pi/6$ gives us $-5\pi/6$ , which is in the range of the inverse sine function. So,
$\sin^{-1}\left(\frac{-1}{2}\right) = -\frac{5\pi}{6}$

STEP 8

Substituting this back into our original problem, we find that $\csc^{-1}(-2) = -\frac{5\pi}{6}$ So, the exact value of the expression $\csc^{-1}(-2)$ is $-\frac{5\pi}{6}$ .

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