To find the exact value of [tex]\(\cos \left[\sin ^{-1}\left(\frac{-4}{5}\right)\right]\)[/tex], follow these steps:
1. Understand the Problem:
We need to find the cosine of an angle whose sine is [tex]\(\frac{-4}{5}\)[/tex].
2. Identify the Given Value:
Let's denote the angle by [tex]\(\theta\)[/tex]. We are given [tex]\(\sin \theta = \frac{-4}{5}\)[/tex].
3. Recall the Fundamental Trigonometric Identity:
Use the Pythagorean identity which states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
4. Substitute the Given Sine Value:
Substitute [tex]\(\sin \theta = \frac{-4}{5}\)[/tex] into the identity:
[tex]\[
\left(\frac{-4}{5}\right)^2 + \cos^2 \theta = 1
\][/tex]
5. Simplify the Equation:
Calculate [tex]\(\left(\frac{-4}{5}\right)^2\)[/tex]:
[tex]\[
\left(\frac{-4}{5}\right)^2 = \frac{16}{25}
\][/tex]
Now, write the equation:
[tex]\[
\frac{16}{25} + \cos^2 \theta = 1
\][/tex]
6. Solve for [tex]\(\cos^2 \theta\)[/tex]:
Rearrange the equation to solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \frac{16}{25}
\][/tex]
Find the common denominator:
[tex]\[
\cos^2 \theta = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
\][/tex]
7. Take the Square Root:
To find [tex]\(\cos \theta\)[/tex], take the square root of both sides. Note that cosine can be positive or negative depending on the quadrant. However, since [tex]\(\sin \theta = \frac{-4}{5}\)[/tex] indicates that [tex]\(\theta\)[/tex] is in the third or fourth quadrant, where cosine is positive in the fourth quadrant and negative in the third quadrant.
Evaluating for [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5}
\][/tex]
8. Determine the Correct Sign:
Knowing that [tex]\(\sin \theta = \frac{-4}{5}\)[/tex] typically places [tex]\(\theta\)[/tex] in the fourth quadrant (negative sine and positive cosine):
[tex]\[
\cos \theta = \frac{3}{5}
\][/tex]
Therefore, the exact value of [tex]\(\cos \left[\sin ^{-1}\left(\frac{-4}{5}\right)\right]\)[/tex] is:
[tex]\[
\boxed{0.6}
\][/tex]
