Question

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

Explanation:

Step1: Recall cotangent definition

$\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{5}{10}$. So the adjacent - side to $\theta$ is 5 and the opposite - side is 10.

Step2: Use Pythagorean theorem

By the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, where $c$ is the hypotenuse, $a = 5$ and $b = 10$. Then $c=\sqrt{5^{2}+10^{2}}=\sqrt{25 + 100}=\sqrt{125}$.

Step3: Recall sine definition

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{10}{\sqrt{125}}=\frac{2}{\sqrt{5}}$ (after simplifying $\frac{10}{\sqrt{125}}=\frac{10}{5\sqrt{5}}=\frac{2}{\sqrt{5}}$). So the number in the square - root is 5.

Answer:

5