Question

25 - 20 college readiness mathematics
right - angle trigonometry
given right triangle def, what is the value of sin(e)?

Explanation:

Step1: Recall sine - ratio definition

The sine of an angle in a right - triangle is defined as $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle E$ in right - triangle $DEF$, the side opposite $\angle E$ is $DF$ and the hypotenuse is $EF$.

Step2: Find the length of $DF$ using the Pythagorean theorem

In right - triangle $DEF$, by the Pythagorean theorem $DF=\sqrt{EF^{2}-DE^{2}}$. Given $DE = 8$ and $EF = 10$, then $DF=\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36}=6$.

Step3: Calculate $\sin(E)$

$\sin(E)=\frac{DF}{EF}$. Substituting $DF = 6$ and $EF = 10$, we get $\sin(E)=\frac{6}{10}=\frac{3}{5}$.

Answer:

$\frac{3}{5}$