2) Find the remaining sides of the right triangle if \( \sin A=\frac{1}{4} \) and \( c=2 \) a) \( a=\frac{1}{2}, b=\frac{\sqrt{15}}{2} \)

To find the remaining sides of the right triangle given that \( \sin A = \frac{1}{4} \) and the hypotenuse \( c = 2 \), we can use the definitions of sine and the Pythagorean theorem. 1. **Using the sine definition**: \[ \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} \] Given \( \sin A = \frac{1}{4} \) and \( c = 2 \): \[ \frac{a}{2} = \frac{1}{4} \] Multiplying both sides by 2: \[ a = \frac{1}{2} \] 2. **Finding the other side \( b \)**: We can use the Pythagorean theorem, which states: \[ a^2 + b^2 = c^2 \] Substituting the known values: \[ \left(\frac{1}{2}\right)^2 + b^2 = 2^2 \] This simplifies to: \[ \frac{1}{4} + b^2 = 4 \] Subtracting \( \frac{1}{4} \) from both sides: \[ b^2 = 4 - \frac{1}{4} = \frac{16}{4} - \frac{1}{4} = \frac{15}{4} \] Taking the square root of both sides: \[ b = \sqrt{\frac{15}{4}} = \frac{\sqrt{15}}{2} \] Thus, the remaining sides of the triangle are: \[ a = \frac{1}{2}, \quad b = \frac{\sqrt{15}}{2} \] So the answer is: \[ a = \frac{1}{2}, \quad b = \frac{\sqrt{15}}{2} \]