To find [tex]\(\sin \theta\)[/tex] given [tex]\(\cos \theta = \frac{4 \sqrt{2}}{7}\)[/tex], we will use the Pythagorean identity which states that:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
1. Substitute the value of [tex]\(\cos \theta\)[/tex] into the identity:
[tex]\[
\cos \theta = \frac{4 \sqrt{2}}{7}
\][/tex]
[tex]\[
\cos^2(\theta) = \left(\frac{4 \sqrt{2}}{7}\right)^2
\][/tex]
2. Calculate [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = \frac{(4 \sqrt{2})^2}{7^2} = \frac{16 \times 2}{49} = \frac{32}{49}
\][/tex]
3. Apply the Pythagorean identity:
[tex]\[
\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \frac{32}{49}
\][/tex]
4. Simplify the expression:
[tex]\[
\sin^2(\theta) = \frac{49}{49} - \frac{32}{49} = \frac{17}{49}
\][/tex]
5. Take the square root to find [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin(\theta) = \sqrt{\sin^2(\theta)} = \sqrt{\frac{17}{49}} = \frac{\sqrt{17}}{7}
\][/tex]
Since the question asks for [tex]\(\sin \theta\)[/tex], we typically take the positive value (unless specified otherwise by a given context or quadrant):
[tex]\[
\sin \theta = \frac{\sqrt{17}}{7}
\][/tex]
Therefore, the value of [tex]\(\sin \theta\)[/tex] is:
[tex]\[
\sin \theta = \frac{\sqrt{17}}{7}
\][/tex]
So, in the form requested:
[tex]\[
\sin \theta = \frac{\sqrt{17}}{7}
\][/tex]
Here, [tex]\( \sin \theta = \frac{\sqrt{17}}{7} \)[/tex].
