QuestionFind the value of $\sec \left[\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ .

Studdy Solution

STEP 1

Assumptions1. The function $\sec \left[\sin ^{-1}\left(-\frac{\sqrt{3}}{}\right)\right]$ is a composition of the secant function and the inverse sine function. . The inverse sine function, $\sin^{-1}$ , gives an angle whose sine is the input value.
3. The secant function, $\sec$ , is the reciprocal of the cosine function.

STEP 2

First, we need to find the angle whose sine is $-\frac{\sqrt{}}{2}$ . This is done by applying the inverse sine function.
$\sin^{-1}\left(-\frac{\sqrt{}}{2}\right)$

STEP 3

The value $-\frac{\sqrt{3}}{2}$ is the sine of $-\frac{\pi}{3}$ or $-\frac{180}{3} = -60$ degrees in the unit circle. So,
$\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}$

STEP 4

Now, we need to find the secant of this angle. The secant is the reciprocal of the cosine, so we need to find the cosine of $-\frac{\pi}{3}$ .
$\cos\left(-\frac{\pi}{3}\right)$

STEP 5

The cosine of $-\frac{\pi}{3}$ is $\frac{1}{2}$ , so $\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}$

STEP 6

Finally, we find the secant of $-\frac{\pi}{3}$ by taking the reciprocal of the cosine.
$\sec\left(-\frac{\pi}{3}\right) = \frac{1}{\cos\left(-\frac{\pi}{3}\right)}$

STEP 7

Substitute the value of $\cos\left(-\frac{\pi}{3}\right)$ into the equation.
$\sec\left(-\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}}$

STEP 8

Calculate the value of the secant.
$\sec\left(-\frac{\pi}{3}\right) =2$ So, the exact value of the expression $\sec \left[\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is2.

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