### Video Transcript

What is the length of side ๐ of
the triangle shown?

Looking at this triangle, we see
that two of the interior angles are given to us. Along with that, the side length
corresponding to one of these interior angles is also known. We also see side length ๐, which
is opposite the angle marked out as 47 degrees. And itโs this length we want to
solve for. To do this, weโre going to use the
sine rule, also known as the law of sines. This rule says that if we have a
triangle, and it could be any triangle, then the ratio of the sine of any one of the
triangleโs interior angles to its corresponding side length is equal to that same
ratio for any of the other angles and corresponding sides.

Now, we donโt need to know all
three of the angles or all three of the sides to use any one of these relations. Because of this, we can apply the
sine rule to our triangle over here in order to solve for side length ๐. We see that the side length, as we
noted, is opposite the interior angle of 47 degrees. So then, as one of our ratios
applying the sine rule, we can write the sin of 47 degrees divided by ๐. And then by that same rule, this
ratio is equal to the sine of any other interior angle in our triangle divided by
its corresponding side length. The question is, do we have that
information?

Well, we see that we have this
other interior angle, 95 degrees, and that opposite that angle we indeed have a
known side length, 10 centimeters. So our ratio then is the sin of 95
degrees divided by 10 centimeters. And as we said, the sine rule tells
us this is equal to the sin of 47 degrees divided by ๐.

And now that we have this equation,
all we need to do is rearrange algebraically to solve for ๐. If we multiply both sides by ๐,
that factor cancels out on the left. And then, next, we multiply both
sides of the equation by 10 centimeters divided by the sin of 95 degrees, canceling
out 10 centimeters and that sine on the right. And finally, we have an expression
that we can evaluate to solve for side length ๐. Entering this expression on our
calculator, to two significant figures, we find a result of 7.3 centimeters. Thatโs the length of side ๐ in the
triangle in our diagram.