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.
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.
\[ \theta = \arcsin\left(\frac{5}{13}\right) \]
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.
Using a calculator:
Calculate \( \theta = \arcsin\left(\frac{5}{13}\right) \approx 0.39479 \) radians.
Calculate \( X \):
\[ X = \pi + 0.39479 \approx 3.53638 \]
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!
