Question

given right triangle def, what is the value of sin(e)?

Explanation:

Step1: Recall sine - ratio definition

In a right - triangle, $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle E$ in right - triangle $DEF$, the side opposite to $\angle E$ is $DF$ and the hypotenuse is $EF$.

Step2: Identify side lengths

We are given that $DF$ (opposite to $\angle E$) and $EF = 10$ (hypotenuse). First, we need to find the length of $DF$ using the Pythagorean theorem. Let $DE = 8$ and $EF=10$. By the Pythagorean theorem $DF=\sqrt{EF^{2}-DE^{2}}=\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36}=6$.

Step3: Calculate $\sin(E)$

Now, $\sin(E)=\frac{DF}{EF}$. Since $DF = 6$ and $EF = 10$, then $\sin(E)=\frac{6}{10}=\frac{3}{5}$.

Answer:

$\frac{3}{5}$