Question

find the cosine of ∠c. write your answer in simplified, rationalized form. do not round. cos(c) =

Explanation:

Step1: Recall cosine formula in right - triangle

In right - triangle $\triangle EDC$ with right - angle at $D$, $\cos(C)=\frac{adjacent}{hypotenuse}$. The side adjacent to $\angle C$ is $DC$ and the hypotenuse is $EC$. First, find the length of $DC$ using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = EC = 3\sqrt{2}$ and $a=ED = 2\sqrt{3}$. Let $b = DC$. Then $b=\sqrt{c^{2}-a^{2}}$.

Step2: Calculate the length of $DC$

$DC=\sqrt{(3\sqrt{2})^{2}-(2\sqrt{3})^{2}}=\sqrt{18 - 12}=\sqrt{6}$.

Step3: Calculate $\cos(C)$

$\cos(C)=\frac{DC}{EC}=\frac{\sqrt{6}}{3\sqrt{2}}$. Rationalize the denominator: $\frac{\sqrt{6}}{3\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{12}}{6}=\frac{2\sqrt{3}}{6}=\frac{\sqrt{3}}{3}$.

Answer:

$\frac{\sqrt{3}}{3}$