Question

given right triangle def, what is the value of sin(e)?
options:
\\(\frac{3}{5}\\)
\\(\frac{3}{4}\\)
\\(\frac{4}{5}\\)
\\(\frac{4}{3}\\)

Explanation:

Step1: Find the length of side \( DF \)

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

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

Step2: Recall the sine formula

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

$$ \sin(E)=\frac{DF}{EF}$$

Step3: Calculate \( \sin(E) \)

Substitute \( DF = 6 \) and \( EF = 10 \) into the formula.

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

Answer:

\(\frac{3}{5}\) (the first option)