Question

given right triangle def, what is the value of sin(e)?
3/5
3/4
4/5
4/3

Explanation:

Step1: Find the length of $DF$

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c = EF = 10\), \(a=DE = 8\), and \(b = DF\)).

$$DF=\sqrt{EF^{2}-DE^{2}}=\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36}=6$$

Step2: Recall the sine function definition

For an angle \(E\) in a right - triangle, \(\sin(E)=\frac{\text{opposite}}{\text{hypotenuse}}\).
The side opposite to angle \(E\) is \(DF = 6\), and the hypotenuse \(EF=10\).

$$ \sin(E)=\frac{DF}{EF}=\frac{6}{10}=\frac{3}{5} $$

Answer:

$\frac{3}{5}$