Question

17 in right triangle abc, ab is the hypotenuse. if ac = 5 and bc = 12, express \frac{\sin b}{\tan b} as a fraction in lowest terms.

Explanation:

Step1: Calculate the hypotenuse \(AB\) using Pythagorean theorem

In right - triangle \(ABC\) with \(AC = 5\) and \(BC=12\), by \(AB=\sqrt{AC^{2}+BC^{2}}\), we have \(AB=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\).

Step2: Find \(\sin B\) and \(\tan B\)

By the definition of sine (\(\sin B=\frac{AC}{AB}\)) and tangent (\(\tan B=\frac{AC}{BC}\)) in a right - triangle.
\(\sin B=\frac{AC}{AB}=\frac{5}{13}\), \(\tan B=\frac{AC}{BC}=\frac{5}{12}\).

Step3: Calculate \(\frac{\sin B}{\tan B}\)

\(\frac{\sin B}{\tan B}=\frac{\frac{5}{13}}{\frac{5}{12}}=\frac{5}{13}\times\frac{12}{5}\) (using the rule \(\frac{a/b}{c/d}=\frac{a}{b}\times\frac{d}{c}\)).

Answer:

\(\frac{12}{13}\)