If \(\sin \theta = \frac{3}{5}\), find \(\cos \theta\).
\(\frac{4}{5}\)
We are given that \(\sin \theta = \frac{3}{5}\). We know that \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\) in a right-angled triangle. Therefore, we can construct a right-angled triangle where the opposite side has length 3 and the hypotenuse has length 5.
Using the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse, we can find the length of the adjacent side:
\(3^2 + b^2 = 5^2\)
\(9 + b^2 = 25\)
\(b^2 = 25 - 9\)
\(b^2 = 16\)
\(b = 4\)
Now we can find \(\cos \theta\). We know that \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\). From our triangle, the adjacent side has length 4 and the hypotenuse has length 5. Therefore:
\(\cos \theta = \frac{4}{5}\)
Therefore, the correct answer is \(\frac{4}{5}\).
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