Calculate the value of cosalpha if sinalpha =-(24)/(25) and alpha in (pi ;3pi )

To calculate the value of $\cos\alpha$ given that $\sin\alpha = -\frac{24}{25}$ and $\alpha \in (\pi, 3\pi)$, we can use the Pythagorean identity:<br /><br />$\sin^2\alpha + \cos^2\alpha = 1$<br /><br />Substituting the given value of $\sin\alpha$, we have:<br /><br />$\left(-\frac{24}{25}\right)^2 + \cos^2\alpha = 1$<br /><br />Simplifying the equation, we get:<br /><br />$\frac{576}{625} + \cos^2\alpha = 1$<br /><br />$\cos^2\alpha = 1 - \frac{576}{625}$<br /><br />$\cos^2\alpha = \frac{49}{625}$<br /><br />Since $\alpha \in (\pi, 3\pi)$, $\cos\alpha$ is negative. Therefore, we take the negative square root:<br /><br />$\cos\alpha = -\sqrt{\frac{49}{625}}$<br /><br />$\cos\alpha = -\frac{7}{25}$<br /><br />Therefore, the value of $\cos\alpha$ is $-\frac{7}{25}$.