Question

find the sine of ∠s.
right triangle rqs with right angle at r, rq = 8, rs = 2√5
write your answer in simplified, rationalized form. do not round.
sin(s) = blank box fraction and square root buttons

Explanation:

Step1: Recall sine definition in right triangle

In a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle S$, the opposite side to $\angle S$ is $RQ = 8$, and we need to find the hypotenuse $SQ$ first.

Step2: Calculate hypotenuse using Pythagorean theorem

The Pythagorean theorem is $a^2 + b^2 = c^2$, where $a = 2\sqrt{5}$, $b = 8$, and $c = SQ$.
So, $SQ^2=(2\sqrt{5})^2 + 8^2 = 20 + 64 = 84$. Then $SQ=\sqrt{84}=\sqrt{4\times21}=2\sqrt{21}$.

Step3: Calculate $\sin(S)$

Using the sine definition, $\sin(S)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{8}{2\sqrt{21}}$. Simplify this fraction: $\frac{8}{2\sqrt{21}}=\frac{4}{\sqrt{21}}$. Rationalize the denominator by multiplying numerator and denominator by $\sqrt{21}$: $\frac{4\sqrt{21}}{21}$.

Answer:

$\frac{4\sqrt{21}}{21}$