Sure! Let's find [tex]\(\cos \theta\)[/tex] given that [tex]\(\sin \theta = \frac{15}{17}\)[/tex] and [tex]\(0 < \theta < \frac{\pi}{2}\)[/tex].
1. Recall the Pythagorean Identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
2. Substitute the given [tex]\(\sin \theta\)[/tex] value:
[tex]\[
\left(\frac{15}{17}\right)^2 + \cos^2 \theta = 1
\][/tex]
3. Square the sine value:
[tex]\[
\left(\frac{15}{17}\right)^2 = \frac{225}{289}
\][/tex]
4. Substitute this value into the identity:
[tex]\[
\frac{225}{289} + \cos^2 \theta = 1
\][/tex]
5. Isolate [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \frac{225}{289}
\][/tex]
6. Find a common denominator and subtract:
[tex]\[
1 = \frac{289}{289} \implies \frac{289}{289} - \frac{225}{289} = \frac{64}{289}
\][/tex]
Thus,
[tex]\[
\cos^2 \theta = \frac{64}{289}
\][/tex]
7. Take the square root of both sides to solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \sqrt{\frac{64}{289}} = \frac{\sqrt{64}}{\sqrt{289}} = \frac{8}{17}
\][/tex]
8. Since [tex]\(0 < \theta < \frac{\pi}{2}\)[/tex], [tex]\(\cos \theta\)[/tex] is positive, making
[tex]\[
\cos \theta = \frac{8}{17}
\][/tex]
Given the values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex], we can also express them in decimal form for completeness:
[tex]\[
\sin \theta \approx 0.8824
\][/tex]
[tex]\[
\cos \theta \approx 0.4706
\][/tex]
Therefore, the values are [tex]\(\sin \theta = \frac{15}{17}\)[/tex] and [tex]\(\cos \theta = \frac{8}{17}\)[/tex], which in decimal form are approximately [tex]\(0.8824\)[/tex] and [tex]\(0.4706\)[/tex], respectively.
