Question

the exact value of \\(sin \theta\\) in simplest radical form.

(there is a right triangle bcd with right angle at c. the hypotenuse bd is \\(sqrt{96}\\), the side bc is \\(sqrt{24}\\), and the side cd is 8?)

attempt 1 out of 3

answer =
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Explanation:

Step1: Recall sine definition for right triangles

$\sin B = \frac{\text{Opposite to } B}{\text{Hypotenuse}}$

Step2: Identify opposite side to B

Opposite side is $AC=8$, hypotenuse is $AB=\sqrt{75}$

Step3: Substitute values into formula

$\sin B = \frac{8}{\sqrt{75}}$

Step4: Rationalize the denominator

$\sin B = \frac{8\sqrt{75}}{75} = \frac{8 \times 5\sqrt{3}}{75} = \frac{40\sqrt{3}}{75}$

Step5: Simplify the fraction

$\sin B = \frac{8\sqrt{3}}{15}$

Answer:

$\frac{8\sqrt{3}}{15}$