Question

find the exact value of sin l in simplest radical form.

Explanation:

Step1: Recall the definition of sine in a right - triangle

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle L\), the side opposite to \(\angle L\) is \(KJ = 7\), and the hypotenuse is \(JL\).

Step2: Calculate \(\sin L\)

We know that \(\sin L=\frac{KJ}{JL}\). Given \(KJ = 7\) and \(JL=\sqrt{56}\), then \(\sin L=\frac{7}{\sqrt{56}}\).
Simplify \(\frac{7}{\sqrt{56}}\):
First, simplify \(\sqrt{56}=\sqrt{4\times14} = 2\sqrt{14}\).
So \(\frac{7}{\sqrt{56}}=\frac{7}{2\sqrt{14}}\).
Rationalize the denominator: \(\frac{7}{2\sqrt{14}}\times\frac{\sqrt{14}}{\sqrt{14}}=\frac{7\sqrt{14}}{2\times14}=\frac{\sqrt{14}}{4}\).

Answer:

\(\frac{\sqrt{14}}{4}\)