To find the missing side of the right triangle, we'll use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the length of the hypotenuse (longest side), and \( a \) and \( b \) are the lengths of the other two sides. Here, we know that one side \( a = 12 \) and the hypotenuse \( c = 20 \).
First, we set up the equation:
\[ 12^2 + b^2 = 20^2 \]
Calculating the squares:
\[ 144 + b^2 = 400 \]
Now, isolate \( b^2 \):
\[ b^2 = 400 - 144 \] \[ b^2 = 256 \]
Now, take the square root:
\[ b = \sqrt{256} = 16 \]
So, the missing side (the other leg) is 16.
Next, we need to find the sine of both angles (let's call the angles opposite sides 12 and 16, respectively).
- For the angle opposite the side of length 12 (let's call it \( \theta \)):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{20} = \frac{3}{5} \]
- For the angle opposite the side of length 16 (let's call it \( \phi \)):
\[ \sin(\phi) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{20} = \frac{4}{5} \]
So we summarize:
- The missing side is 16.
- The sine of the angle opposite the side of length 12 is \(\frac{3}{5}\).
- The sine of the other non-right angle is \(\frac{4}{5}\).