To solve for [tex]\(\cos(x)\)[/tex] given that [tex]\(\sin(x) = 0.9\)[/tex], we can use the Pythagorean identity in trigonometry, which states:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
1. We start by substituting the given value of [tex]\(\sin(x)\)[/tex] into the identity. First, calculate [tex]\(\sin^2(x)\)[/tex]:
[tex]\[
\sin(x) = 0.9 \implies \sin^2(x) = (0.9)^2 = 0.81
\][/tex]
2. Next, we substitute [tex]\(\sin^2(x)\)[/tex] into the Pythagorean identity to solve for [tex]\(\cos^2(x)\)[/tex]:
[tex]\[
0.81 + \cos^2(x) = 1
\][/tex]
[tex]\[
\cos^2(x) = 1 - 0.81 = 0.19
\][/tex]
3. To find [tex]\(\cos(x)\)[/tex], take the square root of [tex]\(\cos^2(x)\)[/tex]:
[tex]\[
\cos(x) = \sqrt{0.19}
\][/tex]
4. Calculating the square root gives us:
[tex]\[
\cos(x) \approx 0.4358898943540673
\][/tex]
5. Finally, round this value to the nearest hundredth:
[tex]\[
\cos(x) \approx 0.44
\][/tex]
Thus, the value of [tex]\(\cos(x)\)[/tex] rounded to the nearest hundredth is [tex]\(0.44\)[/tex].
