Find the exact value of the indicated trigonometric function, using the given information. \( \sin \theta \) if \( \cos \theta=\frac{3}{5} \); terminal side of \( \theta \) in Quadrant । \( \sin \theta=\square \)

To find the value of \( \sin \theta \) given that \( \cos \theta = \frac{3}{5} \) and that the terminal side of \( \theta \) is in Quadrant I, we can use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] ### Step 1: Substitute the known value of \( \cos \theta \) We know that \( \cos \theta = \frac{3}{5} \). Therefore, we can substitute this value into the Pythagorean identity: \[ \sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1 \] ### Step 2: Calculate \( \left(\frac{3}{5}\right)^2 \) Calculating \( \left(\frac{3}{5}\right)^2 \): \[ \left(\frac{3}{5}\right)^2 = \frac{9}{25} \] ### Step 3: Substitute and solve for \( \sin^2 \theta \) Now we substitute this back into the equation: \[ \sin^2 \theta + \frac{9}{25} = 1 \] To isolate \( \sin^2 \theta \), we subtract \( \frac{9}{25} \) from both sides: \[ \sin^2 \theta = 1 - \frac{9}{25} \] ### Step 4: Convert 1 to a fraction with a denominator of 25 We can express 1 as \( \frac{25}{25} \): \[ \sin^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \] ### Step 5: Take the square root Now, we take the square root of both sides to find \( \sin \theta \): \[ \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \] ### Step 6: Determine the sign of \( \sin \theta \) Since \( \theta \) is in Quadrant I, where sine is positive, we have: \[ \sin \theta = \frac{4}{5} \] Thus, the exact value of \( \sin \theta \) is: \[ \sin \theta = \frac{4}{5} \]