A triangle has two angles that measures
110 degrees and 40 degrees. If the side
opposite the 110 degrees angle measures
12 , find the length of the side opposite
40 degree angle. round your answer
to nearest tenth.

To find the length of the side opposite the 40-degree angle in the triangle, we can use the Law of Sines. According to the Law of Sines, the ratio of the length of a side to the sine of its opposite angle is constant for all sides of the triangle. In this case, we have:

Where:

Now we can set up the equation:

Solving for :

Using a calculator:

and

Substituting these values in:

So, the length of the side opposite the 40-degree angle is approximately 8.2 when rounded to the nearest tenth!

Did you know that a triangle can never have more than 180 degrees in total? This is due to the fact that the sum of angles in any triangle is always 180 degrees. In this case, the third angle can be found by subtracting the sum of the two given angles from 180 degrees!

Another fun fact: the Law of Sines can also be used not just in triangles, but in various fields like architecture, aeronautics, and even animation! Understanding how these angles and sides relate helps professionals create stable structures, design efficient flight paths, and produce realistic motion in animated films. Keep this handy for real-world problem-solving!