Question

find the tangent of ∠u. write your answer in simplified, rationalized form. do not round. tan(u) =

Explanation:

Step1: Recall tangent formula

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle U$, the opposite side to $\angle U$ is $TV = \sqrt{85}$ and the adjacent side is $UV$.

Step2: Use Pythagorean theorem to find $UV$

Let $UV = x$. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 3\sqrt{11}$ and $a=\sqrt{85}$. So $x^{2}+(\sqrt{85})^{2}=(3\sqrt{11})^{2}$. Then $x^{2}+85 = 9\times11=99$, and $x^{2}=99 - 85=14$, so $x=\sqrt{14}$.

Step3: Calculate $\tan(U)$

$\tan(U)=\frac{\text{opposite}}{\text{adjacent}}=\frac{\sqrt{85}}{\sqrt{14}}$. Rationalize the denominator: $\tan(U)=\frac{\sqrt{85}\times\sqrt{14}}{\sqrt{14}\times\sqrt{14}}=\frac{\sqrt{1190}}{14}$.

Answer:

$\frac{\sqrt{1190}}{14}$