Question

find the sine of $\angle t$.

write your answer in simplified, rationalized form. do not round.
$\sin(t) = \square$

Explanation:

Step1: Identify hypotenuse length

First, calculate the hypotenuse $TS$ using the Pythagorean theorem:
$$TS = \sqrt{(2\sqrt{7})^2 + 7^2} = \sqrt{28 + 49} = \sqrt{77}$$

Step2: Define sine of $\angle T$

Sine is opposite over hypotenuse. The side opposite $\angle T$ is $US = 2\sqrt{7}$:
$$\sin(T) = \frac{2\sqrt{7}}{\sqrt{77}}$$

Step3: Rationalize and simplify

Factor $\sqrt{77} = \sqrt{11 \times 7}$, then rationalize:
$$\sin(T) = \frac{2\sqrt{7}}{\sqrt{7}\times\sqrt{11}} = \frac{2}{\sqrt{11}} = \frac{2\sqrt{11}}{11}$$

Answer:

$\frac{2\sqrt{11}}{11}$