Video Transcript
𝐴𝐵𝐶 is a triangle, where the
measure of the angle 𝐴 is 138 degrees, 𝑎 is 13 centimeters, and 𝑏 is seven
centimeters. Find the measure of the angle at
𝐵, giving the answer to the nearest second.
Let’s begin by sketching a diagram
of this triangle. Remember the diagram doesn’t 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. We’ll need to use the law of sines
to find the measure of the angle at 𝐵. 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 𝐶, which we don’t know. The law of sines says that sin 𝐴
over 𝑎 equals sin 𝐵 over 𝑏 which equals sin 𝐶 over 𝑐. Alternatively, we could write that
as 𝑎 over sin 𝐴, which equals 𝑏 over sin 𝐵, which equals 𝑐 over sin 𝐶. We can use either of these
formulae. In fact, we choose to use the first
formula because it will minimize the amount of rearranging we’ll need to do to solve
our equation.
The second formula is more useful
when you’re 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 𝐴 and the length of the side 𝑎. We also know the measure of the
angle at 𝐵, and we know the length of the side 𝑏. Since we don’t know anything about
the side or the angle 𝐶, we’ll use sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏.
Substituting the values from our
triangle into the formula gives us sine of 138 degrees over 13 is equal to sine of
𝑥 over seven. To solve, we’ll multiply both sides
by seven. That gives us sine of 𝑥 is equal
to seven sine of 138 degrees all over 13. That’s equal to 0.360. Next, we’ll find the inverse sine
of both sides of this equation. That gives us that 𝑥 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. That’s base 60. We know the integer part of this
number tells us that there are 21 degrees. So let’s deal with the decimal
part. If we subtract 21 from our original
number, we’re left with purely the decimal part. That’s 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. Let’s then deal with the decimal
part of 7.121. If we subtract seven from this
decimal, we’re 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 it’s 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 𝐵
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. That’ll save us a little bit of
time.
