To find [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{5}{13}\)[/tex], we can use the Pythagorean identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
First, let's square the given [tex]\(\cos(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = \left(\frac{5}{13}\right)^2 = \frac{25}{169}
\][/tex]
Then we use the Pythagorean identity to find [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \frac{25}{169}
\][/tex]
Next, we need a common denominator to subtract the fractions:
[tex]\[
1 = \frac{169}{169} \quad \text{so} \quad 1 - \frac{25}{169} = \frac{169}{169} - \frac{25}{169} = \frac{144}{169}
\][/tex]
Now, we take the square root of [tex]\(\sin^2(\theta)\)[/tex] to find [tex]\(\sin(\theta)\)[/tex]:
[tex]\[
\sin(\theta) = \sqrt{\frac{144}{169}} = \frac{\sqrt{144}}{\sqrt{169}} = \frac{12}{13}
\][/tex]
Since we assume [tex]\(\theta\)[/tex] is in the first quadrant (where both sine and cosine are positive), we take the positive root.
Therefore,
[tex]\[
\sin(\theta) = \frac{12}{13}
\][/tex]
