Question

find the sine, cosine, and tangent of ∠s.
write your answer in simplified, rationalized form. do not round.
sin (s) =
cos (s) =
tan (s) =

Explanation:

Step1: Find side length $TR$

Use Pythagorean theorem: $TR = \sqrt{ST^2 - SR^2}$
$TR = \sqrt{(2\sqrt{17})^2 - 6^2} = \sqrt{68 - 36} = \sqrt{32} = 4\sqrt{2}$

Step2: Calculate $\sin(S)$

Opposite over hypotenuse: $\sin(S) = \frac{TR}{ST}$
$\sin(S) = \frac{4\sqrt{2}}{2\sqrt{17}} = \frac{2\sqrt{2}}{\sqrt{17}} = \frac{2\sqrt{34}}{17}$

Step3: Calculate $\cos(S)$

Adjacent over hypotenuse: $\cos(S) = \frac{SR}{ST}$
$\cos(S) = \frac{6}{2\sqrt{17}} = \frac{3}{\sqrt{17}} = \frac{3\sqrt{17}}{17}$

Step4: Calculate $\tan(S)$

Opposite over adjacent: $\tan(S) = \frac{TR}{SR}$
$\tan(S) = \frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3}$

Answer:

$\sin(S) = \frac{2\sqrt{34}}{17}$
$\cos(S) = \frac{3\sqrt{17}}{17}$
$\tan(S) = \frac{2\sqrt{2}}{3}$