Question

if $cot \theta=\frac{5}{10}$, what is $sin \theta$? $sin \theta=\frac{2}{sqrt{?}}$

Explanation:

Step1: Recall the definition of cotangent

Cotangent is defined as \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\). Given \(\cot\theta = \frac{5}{10}\), we can consider the adjacent side \(a = 5\) and the opposite side \(o=10\).

Step2: Use the Pythagorean theorem to find the hypotenuse

The Pythagorean theorem is \(h=\sqrt{a^{2}+o^{2}}\), where \(h\) is the hypotenuse. Substituting \(a = 5\) and \(o = 10\), we get \(h=\sqrt{5^{2}+10^{2}}=\sqrt{25 + 100}=\sqrt{125}\).

Step3: Recall the definition of sine

Sine is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\sin\theta=\frac{10}{\sqrt{125}}\), and simplifying \(\frac{10}{\sqrt{125}}=\frac{2}{\sqrt{5}}\) (dividing numerator and denominator by \(5\)).

Answer:

\(5\)