9. Solve for the required ratios in each of the following Express each answer as an exact value with a rational denominator. a) Angle Theta terminates in the second quadrant. If tanTheta =-(sqrt (3))/(5) find sinTheta and cosTheta

## Step 1: Find the value of $r$<br />### Given that $\tan(\theta) = -\frac{\sqrt{3}}{5}$, we know that $\tan(\theta) = \frac{y}{x}$. Since $\theta$ is in the second quadrant, $x$ is negative and $y$ is positive. Thus, $x = -5$ and $y = \sqrt{3}$. We can find $r$ using the Pythagorean theorem: $r^2 = x^2 + y^2$. $r^2 = (-5)^2 + (\sqrt{3})^2 = 25 + 3 = 28$. Therefore, $r = \sqrt{28} = 2\sqrt{7}$.<br /><br />## Step 2: Calculate $\sin(\theta)$<br />### Now that we have $y$ and $r$, we can find $\sin(\theta)$. $\sin(\theta) = \frac{y}{r} = \frac{\sqrt{3}}{2\sqrt{7}}$. Rationalizing the denominator gives $\sin(\theta) = \frac{\sqrt{21}}{14}$.<br /><br />## Step 3: Calculate $\cos(\theta)$<br />### Similarly, we can find $\cos(\theta)$ using $x$ and $r$. $\cos(\theta) = \frac{x}{r} = \frac{-5}{2\sqrt{7}}$. Rationalizing the denominator gives $\cos(\theta) = -\frac{5\sqrt{7}}{14}$.