Calculate the value of cosalpha if sinalpha =-(24)/(25) and alpha in (pi ;3pi /2):

To calculate the value of $\cos\alpha$ given that $\sin\alpha = -\frac{24}{25}$ and $\alpha \in (\pi, 3\pi/2)$, 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 />$\cos^2\alpha = \frac{625}{625} - \frac{576}{625}$<br /><br />$\cos^2\alpha = \frac{49}{625}$<br /><br />Taking the square root of both sides, we have:<br /><br />$\cos\alpha = \pm \frac{7}{25}$<br /><br />Since $\alpha \in (\pi, 3\pi/2)$, $\cos\alpha$ is negative. Therefore, the value of $\cos\alpha$ is:<br /><br />$\cos\alpha = -\frac{7}{25}$