Question

if $csc \theta=\frac{sqrt{125}}{10}$, what is $\tan \theta$?
$\tan \theta=?$

Explanation:

Step1: Recall the definition of \(\csc\theta\)

\(\csc\theta=\frac{\text{Hypotenuse}}{\text{Opposite}}\). Given \(\csc\theta = \frac{\sqrt{125}}{10}\), so Hypotenuse \(=\sqrt{125}\), Opposite \( = 10\).

Step2: Use the Pythagorean theorem to find the adjacent side

Let the adjacent side be \(a\). By \(a^{2}+\text{Opposite}^{2}=\text{Hypotenuse}^{2}\), \(a^{2}+10^{2}=(\sqrt{125})^{2}\).
\(a^{2}+100 = 125\), then \(a^{2}=125 - 100=25\), so \(a = 5\).

Step3: Recall the definition of \(\tan\theta\)

\(\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}\). Substitute Opposite \( = 10\) and Adjacent \(=5\) into the formula.
\(\tan\theta=\frac{10}{5}\)

Answer:

\(2\)