Question

find the sine of ∠i.
write your answer in simplified, rationalized form. do not round.
sin (i) =

Explanation:

Step1: Use Pythagorean theorem to find \(HI\)

In right - triangle \(HGI\), by Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = HG=\sqrt{35}\), \(b = GI=\sqrt{35}\), and \(c = HI\). Then \(HI=\sqrt{(\sqrt{35})^{2}+(\sqrt{35})^{2}}=\sqrt{35 + 35}=\sqrt{70}\)

Step2: Use the definition of sine function

The sine function is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle I\), the opposite side to \(\angle I\) is \(HG=\sqrt{35}\), and the hypotenuse is \(HI=\sqrt{70}\). So \(\sin(I)=\frac{HG}{HI}=\frac{\sqrt{35}}{\sqrt{70}}\)

Step3: Rationalize the denominator

\(\frac{\sqrt{35}}{\sqrt{70}}=\frac{\sqrt{35}}{\sqrt{35\times2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\)

Answer:

\(\frac{\sqrt{2}}{2}\)