To evaluate [tex]\(\sin \left(\operatorname{Arccos}\left(-\frac{4}{\sqrt{25}}\right)\right)\)[/tex], let’s go through the steps carefully.
1. Simplify the argument inside the [tex]\(\operatorname{Arccos}\)[/tex]:
Given [tex]\(-\frac{4}{\sqrt{25}}\)[/tex], we can simplify [tex]\(\sqrt{25}\)[/tex] to 5. So the expression inside [tex]\(\operatorname{Arccos}\)[/tex] becomes:
[tex]\[
-\frac{4}{\sqrt{25}} = -\frac{4}{5} = -0.8.
\][/tex]
2. Identify the trigonometric identity to use:
We use the identity for the sine of the arc-cosine function:
[tex]\[
\sin(\operatorname{Arccos}(x)) = \sqrt{1 - x^2}.
\][/tex]
Here, [tex]\(x = -0.8\)[/tex].
3. Calculate [tex]\(1 - x^2\)[/tex]:
First, calculate [tex]\(x^2\)[/tex]:
[tex]\[
x = -0.8 \implies x^2 = (-0.8)^2 = 0.64.
\][/tex]
Then, calculate [tex]\(1 - x^2\)[/tex]:
[tex]\[
1 - x^2 = 1 - 0.64 = 0.36.
\][/tex]
4. Take the square root of the result:
[tex]\[
\sqrt{0.36} = 0.6.
\][/tex]
So, [tex]\(\sin \left(\operatorname{Arccos}\left(-0.8\right)\right) = 0.6\)[/tex].
5. Validate against the provided choices:
The four options provided are:
- [tex]\(-\frac{\sqrt{25}}{\sqrt{128}}\)[/tex]
- [tex]\(\frac{3}{\sqrt{25}}\)[/tex]
- [tex]\(\frac{\sqrt{25}}{\sqrt{128}}\)[/tex]
- [tex]\(-\frac{3}{\sqrt{25}}\)[/tex]
Let’s check these values:
- [tex]\(\sqrt{25} = 5\)[/tex].
- [tex]\(\frac{3}{\sqrt{25}} = \frac{3}{5} = 0.6\)[/tex].
This matches our calculated result [tex]\(0.6\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{3}{\sqrt{25}}}
\][/tex]
