Question

find the cosine of ∠c. simplify your answer and write it as a proper fraction, improper fraction, or whole number. cos (c) =

Explanation:

Step1: Recall cosine definition

In a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ for an acute angle $\theta$.

Step2: Identify sides for $\angle C$

For $\angle C$, the adjacent side is $CD$ (we need to find its length first using Pythagoras: $CD = \sqrt{CE^2 - DE^2} = \sqrt{52^2 - 48^2} = \sqrt{2704 - 2304} = \sqrt{400} = 20$). The hypotenuse is $CE = 52$, and the adjacent side to $\angle C$ is $CD = 20$? Wait, no—wait, $\angle C$: in triangle $CDE$, right-angled at $D$. So $\angle C$: adjacent side is $CD$, opposite is $DE$, hypotenuse $CE$. Wait, no, let's re-express: in right triangle $CDE$, right angle at $D$. So angle at $C$: sides: adjacent is $CD$, hypotenuse $CE$, opposite is $DE$. Wait, but we found $CD = 20$, $DE = 48$, $CE = 52$. So $\cos(C) = \frac{\text{adjacent to } C}{\text{hypotenuse}} = \frac{CD}{CE} = \frac{20}{52}$? Wait, no, wait, maybe I mixed up. Wait, angle at $C$: the sides: the leg adjacent to $C$ is $CD$, and the hypotenuse is $CE$. Wait, but let's check again. Wait, $CE$ is 52 (the side opposite the right angle? No, wait, right angle is at $D$, so hypotenuse is $CE$, length 52. Then $DE$ is 48 (one leg), $CD$ is the other leg. So by Pythagoras, $CD = \sqrt{52^2 - 48^2} = \sqrt{(52 - 48)(52 + 48)} = \sqrt{4 \times 100} = \sqrt{400} = 20$. So now, for angle $C$: adjacent side (to angle $C$) is $CD = 20$, hypotenuse $CE = 52$. Wait, no—wait, angle $C$: the angle at $C$, so the sides: the side forming angle $C$ with the hypotenuse is $CD$ (adjacent) and $CE$ (hypotenuse). Wait, but maybe I had it backwards. Wait, in angle $C$, the adjacent side is $CD$, and the opposite is $DE$. So $\cos(C) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{CD}{CE} = \frac{20}{52}$? Wait, no, that can't be, because 20 and 52 simplify. Wait, wait, no—wait, maybe I mixed up adjacent and opposite. Wait, angle $C$: let's label the triangle: $C$ is a vertex, $D$ is right angle, $E$ is the other vertex. So sides: $CD$ (from $C$ to $D$), $DE$ (from $D$ to $E$), $CE$ (from $C$ to $E$). So angle at $C$: between $CD$ and $CE$. So the adjacent side is $CD$, hypotenuse is $CE$, so $\cos(C) = \frac{CD}{CE} = \frac{20}{52} = \frac{5}{13}$ when simplified (dividing numerator and denominator by 4: 20 ÷ 4 = 5, 52 ÷ 4 = 13). Wait, yes, 20/52 simplifies to 5/13.

Step3: Simplify the fraction

$\frac{20}{52} = \frac{20 \div 4}{52 \div 4} = \frac{5}{13}$.

Answer:

$\frac{5}{13}$