Question

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

Explanation:

Step1: Identify right triangle sides

In right $\triangle CDE$, $\angle D=90^\circ$.

• Hypotenuse $CE = \sqrt{82}$
• Adjacent to $\angle C$: $CD$ (unknown), Opposite to $\angle C$: $DE=6\sqrt{2}$

Step2: Calculate length of $CD$

Use Pythagorean theorem: $CD^2 + DE^2 = CE^2$

$$\begin{align*} CD^2 + (6\sqrt{2})^2 &= (\sqrt{82})^2\\ CD^2 + 72 &= 82\\ CD^2 &= 82-72=10\\ CD &= \sqrt{10} \end{align*}$$

Step3: Apply cosine definition

Cosine of $\angle C$ is $\frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{CD}{CE}$

$$ \cos(C) = \frac{\sqrt{10}}{\sqrt{82}} $$

Step4: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{82}$:

$$ \cos(C) = \frac{\sqrt{10} \times \sqrt{82}}{\sqrt{82} \times \sqrt{82}} = \frac{\sqrt{820}}{82} = \frac{\sqrt{4 \times 205}}{82} = \frac{2\sqrt{205}}{82} = \frac{\sqrt{205}}{41} $$

Answer:

$\frac{\sqrt{205}}{41}$