Given \(\sin \theta = \frac{3}{5}\) and \(\theta\) is in the first quadrant, determine \(\cos \theta\). In the first quadrant, both sine and cosine are positive. The Pythagorean identity is \(\sin^2 \theta + \cos^2 \theta = 1\).
Substitute \(\sin \theta = \frac{3}{5}\) into the identity:
\(\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1\)
\(\frac{9}{25} + \cos^2 \theta = 1\)
Isolate \(\cos^2 \theta\):
\(\cos^2 \theta = 1 - \frac{9}{25}\)
\(\cos^2 \theta = \frac{25}{25} - \frac{9}{25}\)
\(\cos^2 \theta = \frac{16}{25}\)
Since \(\theta\) is in the first quadrant, take the positive square root:
\(\cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}\)
The result is \(\frac{4}{5}\).
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