### Video Transcript

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

In this diagram, we see side ๐
right here, as well as two interior angles of this triangle, 64 degrees and 38
degrees. And then opposite this angle of 38
degrees, we have a side length given as 6.1 centimeters. Knowing this, we want to solve for
the side length ๐. And weโre going to do it by
applying whatโs called the sine rule. This rule says that if we have a
triangle, and it could be any triangle, with its angles and sides marked out like
this, then the ratio of the sine of any of these angles to the corresponding side
length is equal to that same ratio for the other pairs of sides and angles.

We can see that a key to being able
to use the sine rule is to have at least two corresponding angle and side pairs. That way, we can set up an equality
and then solve for any one of those four values, given the other three. As we look to apply the sine rule
to our diagram in order to solve for the side length ๐, we see that we have one
such angle and side pair, 38 degrees with 6.1 centimeters, but that thatโs the only
one. We donโt have a second
corresponding pair. If we knew what this angle of the
triangle was though, we see that that angle is opposite the side length ๐ we want
to solve for.

It turns out that we can solve for
this angle, but we wonโt do it using the sine rule. Instead, weโll use the fact that
the sum of the interior angles of any triangle is always 180 degrees. So therefore, if we call this
unknown angle in our triangle capital ๐ด, then we can say that ๐ด plus 64 degrees
plus 38 degrees is equal to 180 degrees. Now, 64 degrees plus 38 degrees is
102 degrees. And if we subtract this amount from
the left-hand side, the negative 102 cancels with positive 102. And we find that ๐ด equals 180
degrees minus 102 degrees, or 78 degrees.

Now that we know this angle in our
triangle, we can use the sine rule to solve for the side length ๐. Weโll write that the sin of capital
๐ด, which we know is 78 degrees, divided by the side length lowercase ๐ is equal to
this ratio over here, the sine of this angle in our triangle divided by this side
length. Solving for ๐ is now just a matter
of cross multiplying. We can start by multiplying both
sides of the equation by ๐, leading that factor to cancel on the left. And then, in our resulting
equation, if we want to isolate ๐ on the right-hand side, we do this by multiplying
both sides by the inverse of this ratio. When we multiply this ratio by its
inverse, that product is equal to one, effectively canceling all this out. And finally, we have an expression
we can enter in on our calculator to solve for ๐. To two significant figures, ๐ is
9.7 centimeters. Thatโs the length of this side of
the triangle.