To find the value of [tex]\(\cos \theta\)[/tex] given that [tex]\(\sin \theta = \frac{3}{5}\)[/tex], we can use a trigonometric identity. The Pythagorean identity states that for any angle [tex]\(\theta\)[/tex],
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Given [tex]\(\sin \theta = \frac{3}{5}\)[/tex], we first calculate [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^2 \theta = \left( \frac{3}{5} \right)^2 = \frac{9}{25}
\][/tex]
Next, using the Pythagorean identity, we solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{9}{25}
\][/tex]
Converting 1 to a fraction with a common denominator:
[tex]\[
1 = \frac{25}{25}
\][/tex]
So we have:
[tex]\[
\cos^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
Now, to find [tex]\(\cos \theta\)[/tex], we take the square root of [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}
\][/tex]
Thus, the value of [tex]\(\cos \theta\)[/tex] is [tex]\(\frac{4}{5}\)[/tex].
Reiterating, we calculated:
- [tex]\(\sin^2 \theta = 0.36\)[/tex]
- [tex]\(\cos^2 \theta = 0.64\)[/tex]
- [tex]\(\cos \theta = 0.8\)[/tex]
Thus, [tex]\(\cos \theta = 0.8\)[/tex] or [tex]\(\frac{4}{5}\)[/tex].
