Question

what is the trigonometric ratio for sin c? enter your answer as a fraction in simplest form in the boxes.

Explanation:

Step1: Recall the sine ratio formula

The formula for \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) in a right - triangle. For \(\angle C\), the side opposite to \(\angle C\) is \(AB\), and the hypotenuse is \(AC\). First, find the length of \(AB\) using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 80\), \(c = 82\), and \(b=AB\). So, \(AB=\sqrt{82^{2}-80^{2}}=\sqrt{(82 + 80)(82 - 80)}=\sqrt{162\times2}=\sqrt{324}=18\).

Step2: Calculate \(\sin C\)

Using the sine ratio formula \(\sin C=\frac{\text{opposite}}{\text{hypotenuse}}\). The side opposite to \(\angle C\) is \(AB = 18\) and the hypotenuse \(AC=82\). So, \(\sin C=\frac{18}{82}=\frac{9}{41}\).

Answer:

\(\frac{9}{41}\)