Questions: For the given right triangle, the longer leg is 12 units long and the shorter leg is 8 units long. sin α= (Type an integer or a simplified fraction. Type an exact answer, using radicals as needed. Rationalize all denominators.)

$For the given right triangle, the longer leg is 12 units long and the shorter leg is 8 units long. sin α= (Type an integer or a simplified fraction. Type an exact answer, using radicals as needed. Rationalize all denominators.)$

Transcript text: Question Part 1 of 6 Completed For the given right triangle, the longer leg is 12 units long and the shorter leg is 8 units long. $\boldsymbol{\operatorname { s i n }} \alpha=$ $\square$ (Type an integer or a simplified fraction. Type an exact answer, using radicals as needed. Rationalize all denominators.)

Solution

Solution Steps

To find the sine of angle $\alpha$ in a right triangle, we use the definition of sine, which is the ratio of the length of the opposite side to the hypotenuse. First, we need to calculate the hypotenuse using the Pythagorean theorem. Once we have the hypotenuse, we can find $\sin \alpha$ by dividing the length of the shorter leg (opposite side) by the hypotenuse.

Step 1: Calculate the Hypotenuse

Using the Pythagorean theorem, we find the hypotenuse $ c $ of the right triangle with legs $ a = 8 $ and $ b = 12 $: \[ c = \sqrt{a^2 + b^2} = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} \approx 14.4222 \]

Step 2: Calculate $ \sin \alpha $

The sine of angle $ \alpha $ is given by the ratio of the length of the opposite side (the shorter leg) to the hypotenuse: \[ \sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{\sqrt{208}} \approx 0.5547 \]

Final Answer

Thus, the value of $ \sin \alpha $ is approximately $ 0.5547 $. Therefore, we can express the final answer as: \[ \boxed{\sin \alpha \approx 0.5547} \]

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