To find the value of [tex]\(\cos \left[ \sin^{-1} \left( -\frac{\sqrt{2}}{2} \right) \right]\)[/tex], let's go through a step-by-step solution:
1. Understanding the Expression:
- We need to find [tex]\(\cos(\theta)\)[/tex], where [tex]\(\theta = \sin^{-1}\left( -\frac{\sqrt{2}}{2} \right)\)[/tex].
2. Identify [tex]\(\theta\)[/tex]:
- By definition, [tex]\(\theta = \sin^{-1}\left( -\frac{\sqrt{2}}{2} \right)\)[/tex] implies that [tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex].
3. Determine the Quadrant:
- The range of the inverse sine function, [tex]\(\sin^{-1}\)[/tex], is [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex]. This means [tex]\(\theta\)[/tex] will be in either the first or fourth quadrant. Since [tex]\(\sin(\theta)\)[/tex] is negative, [tex]\(\theta\)[/tex] must lie in the fourth quadrant.
4. Use Pythagorean Identity:
- We know the Pythagorean identity: [tex]\(\sin^2(\theta) + \cos^2(\theta) = 1\)[/tex].
- Substituting [tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex], we get:
[tex]\[
\left( -\frac{\sqrt{2}}{2} \right)^2 + \cos^2(\theta) = 1
\][/tex]
[tex]\[
\frac{2}{4} + \cos^2(\theta) = 1
\][/tex]
[tex]\[
\frac{1}{2} + \cos^2(\theta) = 1
\][/tex]
[tex]\[
\cos^2(\theta) = 1 - \frac{1}{2}
\][/tex]
[tex]\[
\cos^2(\theta) = \frac{1}{2}
\][/tex]
5. Solve for [tex]\(\cos(\theta)\)[/tex]:
- Taking the square root of both sides, we get:
[tex]\[
\cos(\theta) = \pm \frac{\sqrt{2}}{2}
\][/tex]
6. Determine the Sign of Cosine:
- In the fourth quadrant, where [tex]\(\theta\)[/tex] is located, cosine is positive. Therefore:
[tex]\[
\cos(\theta) = \frac{\sqrt{2}}{2}
\][/tex]
7. Conclusion:
- Thus, [tex]\(\cos \left[ \sin^{-1} \left( -\frac{\sqrt{2}}{2} \right) \right] = \frac{\sqrt{2}}{2}\)[/tex].
Given the numerical value calculated, the approximate value for [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is:
[tex]\[
\boxed{0.7071067811865476}
\][/tex]
