SinX= - 5/13, П<х<3п/2

To solve for \( X \) given \( \sin X = -\frac{5}{13} \) in the interval \( \pi < X < \frac{3\pi}{2} \), we need to recognize that the sine function is negative in the third quadrant, which aligns with the given interval. 1. **Finding the reference angle**: First, we find the reference angle \( \theta \) where \( \sin \theta = \frac{5}{13} \). We can find this angle using the arcsine function, but we only need the reference angle since sine is negative in the third quadrant. \[ \theta = \arcsin\left(\frac{5}{13}\right) \] 2. **Calculating \( X \)**: In the third quadrant (where \( \pi < X < \frac{3\pi}{2} \)), the angle \( X \) can be expressed as: \[ X = \pi + \theta \] To find the actual value of \( X \), we will first calculate \( \theta \) using a calculator, keeping in mind some trigonometric values or approximations might be helpful. 3. **Using a calculator**: Calculate \( \theta = \arcsin\left(\frac{5}{13}\right) \approx 0.39479 \) radians. 4. **Calculate \( X \)**: \[ X = \pi + 0.39479 \approx 3.53638 \] 5. **Final answer**: Thus, the solution in the given interval \( \pi < X < \frac{3\pi}{2} \) is \[ X \approx 3.53638 \text{ radians}. \] If you'd like a more precise value or a numerical approximation, feel free to ask!