To evaluate [tex]\(\cos \left(\operatorname{Sin}^{-1} 0\right)\)[/tex], let's break the problem into clear steps:
### Step-by-Step Solution:
1. Identify the Argument:
We need to first understand what [tex]\(\operatorname{Sin}^{-1} 0\)[/tex] (the inverse sine, or arcsine, of 0) means.
2. Evaluate [tex]\(\operatorname{Sin}^{-1} 0\)[/tex]:
By definition, [tex]\(\operatorname{Sin}^{-1} x\)[/tex] is the angle whose sine is [tex]\(x\)[/tex]. Hence, [tex]\(\operatorname{Sin}^{-1} 0\)[/tex] is the angle [tex]\(\theta\)[/tex] such that [tex]\(\sin \theta = 0\)[/tex].
- The sine function is 0 at angles like [tex]\(0, \pi, 2\pi, -\pi, -2\pi\)[/tex], etc.
- However, the principal value range for [tex]\(\operatorname{Sin}^{-1} x\)[/tex] is [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex]. Within this range, the angle [tex]\(\theta\)[/tex] such that [tex]\(\sin \theta = 0\)[/tex] is [tex]\(\theta = 0\)[/tex].
Hence, [tex]\(\operatorname{Sin}^{-1} 0 = 0\)[/tex].
3. Calculate the Cosine:
Now, we need to evaluate [tex]\(\cos (0)\)[/tex].
- Recall that the cosine function of 0 radians is 1, as [tex]\(\cos(0) = 1\)[/tex].
### Conclusion:
Therefore, [tex]\(\cos \left(\operatorname{Sin}^{-1} 0\right) = \cos(0) = 1\)[/tex].
The final result is:
[tex]\[ \cos \left(\operatorname{Sin}^{-1} 0\right) = 1.0 \][/tex]
