### Video Transcript

๐ด๐ต๐ถ is a triangle, where the
measure of the angle ๐ด is 138 degrees, ๐ is 13 centimeters, and ๐ is seven
centimeters. Find the measure of the angle at
๐ต, giving the answer to the nearest second.

Letโs begin by sketching a diagram
of this triangle. Remember the diagram doesnโt need
to be to scale, but it should be roughly in proportion so we can check the validity
of any answers we get. Here we have a nonright-angled
triangle for which we know the length of two sides and one of the angles. Weโll need to use the law of sines
to find the measure of the angle at ๐ต. Remember to use the law of cosines,
we would need a minimum of two sides plus an included angle.

In this triangle, the included
angle would be the angle at ๐ถ, which we donโt know. The law of sines says that sin ๐ด
over ๐ equals sin ๐ต over ๐ which equals sin ๐ถ over ๐. Alternatively, we could write that
as ๐ over sin ๐ด, which equals ๐ over sin ๐ต, which equals ๐ over sin ๐ถ. We can use either of these
formulae. In fact, we choose to use the first
formula because it will minimize the amount of rearranging weโll need to do to solve
our equation.

The second formula is more useful
when youโre trying to calculate the length of a missing side. Remember, for the law of sines, we
need pairs of angles and their associated side. Here we know the measure of the
angle at ๐ด and the length of the side ๐. We also know the measure of the
angle at ๐ต, and we know the length of the side ๐. Since we donโt know anything about
the side or the angle ๐ถ, weโll use sin ๐ด over ๐ equals sin ๐ต over ๐.

Substituting the values from our
triangle into the formula gives us sine of 138 degrees over 13 is equal to sine of
๐ฅ over seven. To solve, weโll multiply both sides
by seven. That gives us sine of ๐ฅ is equal
to seven sine of 138 degrees all over 13. Thatโs equal to 0.360. Next, weโll find the inverse sine
of both sides of this equation. That gives us that ๐ฅ is equal to
the inverse sine of 0.360, which is 21.1186 degrees.

Remember though, the question is
asking us to leave our answer correct to the nearest second. We will need to convert from
degrees into minutes and seconds. Thatโs base 60. We know the integer part of this
number tells us that there are 21 degrees. So letโs deal with the decimal
part. If we subtract 21 from our original
number, weโre left with purely the decimal part. Thatโs 0.118 and so on. We can multiply that by 60 to give
us 7.121.

The integer part of this number
tells us the number of minutes. Currently then, we have 21 degrees
and seven minutes. Letโs then deal with the decimal
part of 7.121. If we subtract seven from this
decimal, weโre left with 0.121 and so on. And we can multiply that by 60 to
tell us the number of seconds. The integer part of this number is
the number of seconds. And since itโs correct to the
nearest second and the deciding number is two, which is less than five, this rounds
down to seven.

The measure of the angle at ๐ต
correct to the nearest second is 21 degrees, seven minutes, and seven seconds. Some calculators do have a button
that will convert an answer into minutes and seconds for us. It looks like this! And simply pressing this button
after calculating the answer of 21.118 and so on will convert it to 21 degrees,
seven minutes, and seven seconds. Thatโll save us a little bit of
time.