6. (1 point) Given that sinTheta =(-3)/(8) and the terminal arm of the angle Theta is located in quadrant 4, determine exact tan ratio of the angle Theta a. (3)/(sqrt (55)) b. (8)/(sqrt (55)) c. (-3)/(sqrt (55)) d. (8)/(sqrt (73))

## Step 1: Finding the cosine value<br />### Since $\sin\Theta = \frac{-3}{8}$ and $\Theta$ is in quadrant 4, we know that $\sin\Theta = \frac{y}{r}$, where $y=-3$ and $r=8$. In quadrant 4, $x$ is positive. We can use the Pythagorean theorem ($x^2 + y^2 = r^2$) to find $x$. $x^2 + (-3)^2 = 8^2 \Rightarrow x^2 + 9 = 64 \Rightarrow x^2 = 55 \Rightarrow x = \sqrt{55}$. Thus, $\cos\Theta = \frac{x}{r} = \frac{\sqrt{55}}{8}$.<br /><br />## Step 2: Calculating the tangent value<br />### Now, we can find $\tan\Theta$ using the identity $\tan\Theta = \frac{\sin\Theta}{\cos\Theta}$. Substituting the values, we get $\tan\Theta = \frac{-3/8}{\sqrt{55}/8} = \frac{-3}{8} \cdot \frac{8}{\sqrt{55}} = \frac{-3}{\sqrt{55}}$.