Certainly! Let's solve for [tex]\(\sin \theta\)[/tex] given that [tex]\(\cos \theta = \frac{8}{17}\)[/tex].
We can use the Pythagorean identity which states:
[tex]\[
\cos^2(\theta) + \sin^2(\theta) = 1
\][/tex]
Given that [tex]\(\cos \theta = \frac{8}{17}\)[/tex], we first need to find [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = \left(\frac{8}{17}\right)^2 = \frac{64}{289}
\][/tex]
Next, we substitute [tex]\(\cos^2(\theta)\)[/tex] into the Pythagorean identity to solve for [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \frac{64}{289}
\][/tex]
To combine the terms, we convert 1 to a fraction with a common denominator of 289:
[tex]\[
1 = \frac{289}{289}
\][/tex]
Now, subtract [tex]\(\frac{64}{289}\)[/tex] from [tex]\(\frac{289}{289}\)[/tex]:
[tex]\[
\sin^2(\theta) = \frac{289}{289} - \frac{64}{289} = \frac{225}{289}
\][/tex]
Therefore, [tex]\(\sin^2(\theta) = \frac{225}{289}\)[/tex].
To find [tex]\(\sin \theta\)[/tex], we take the positive square root of [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin (\theta) = \sqrt{\frac{225}{289}} = \frac{\sqrt{225}}{\sqrt{289}} = \frac{15}{17}
\][/tex]
However, the calculation using a more precise method yields:
[tex]\[
\sin^2(\theta) = 0.7785467128027681
\][/tex]
Taking the positive square root of [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin \theta = \sqrt{0.7785467128027681} = 0.8823529411764706
\][/tex]
So, the correct value for [tex]\(\sin \theta\)[/tex] is approximately:
[tex]\[
\sin \theta = 0.8823529411764706
\][/tex]
