Question

if \\( \cot \theta = \frac { 5 } { 10 } \\), what is \\( \sin \theta \\)? \\( \sin \theta = \frac {? } { \sqrt { \square } } \\)

Explanation:

Step1: Recall the definition of cotangent

$\cot\theta=\frac{\text{adjacent}}{\text{opposite}}$. Given $\cot\theta = \frac{5}{10}$, so the adjacent side is $5$ and the opposite side is $10$.

Step2: Use the Pythagorean theorem to find the hypotenuse

Let the hypotenuse be $c$. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 5$ (adjacent) and $b = 10$ (opposite). Then $c^{2}=5^{2}+10^{2}=25 + 100=125$.

Step3: Recall the definition of sine

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Since the opposite side is $10$ and the hypotenuse $c=\sqrt{125}$, so $\sin\theta=\frac{10}{\sqrt{125}}$.

Answer:

$\frac{10}{\sqrt{125}}$