### Video Transcript

π΄π΅πΆ is a triangle, where π
equals 28 centimeters, π equals 17 centimeters, and the measure of angle πΆ equals
60 degrees. Find the missing length rounded to
three decimal places and the missing angles rounded to the nearest degree.

Letβs start by sketching this
triangle and adding the information we know. Here, we have a triangle, and we
can label it with the vertices π΄, π΅, and πΆ. Conventionally, the side length
opposite the vertex π΄ is labeled with a lowercase π. The side length opposite vertex π΅
would be labeled with the lowercase π. And the side length opposite vertex
πΆ would be labeled with the lowercase π.

We know that π equals 28
centimeters, π equals 17 centimeters, and angle πΆ measures 60 degrees. And we want to solve for the
measures of angle π΄ and π΅. And we want to solve for the
measures of angle π΄ and π΅ and the length of side length π. We know two of the sides and the
angle in between them. We have enough information to use
the law of cosines to solve for that third missing side. The law of cosines tells us that π
squared is equal to π squared plus π squared minus two ππ times cos of π΄.

Letβs think about what this
means. If we know some angle π΄ inside a
triangle and the two side lengths adjacent to angle π΄, we have enough information
to find the side length of the side that is opposite to our angle π΄. In our triangle, weβre actually
missing the side length π and we know angle πΆ. Because we know π΄ and π΅ and angle
πΆ, it might be helpful to rewrite this law in terms of the missing side length
π. Since weβre looking for side length
π, we would say that π squared is equal to π squared plus π squared minus two
ππ times cos of πΆ. And then to solve for π, we just
plug in what we know. π squared will be equal to 28
squared plus 17 squared minus two times 28 times 17 times cos of 60.

Calculating that, we end up with π
squared equals 579. Weβll find π by taking the square
root of both sides. And we find that π equals
24.43358345 continuing. We know that this side length needs
to be rounded to three decimal places. To the right of that three is a
five. And so we can round the side length
π to 24.434 centimeters. And weβll add that value to our
graph.

At this point, we want to find a
missing angle in this triangle. If we look closely, weβll see that
we can rearrange the law of cosines to use it to solve for a missing angle if we
know all three side lengths in a triangle. If we knew side lengths π, π, and
π, the only variable remaining is that opposite angle π΄. Itβs possible to go ahead and plug
in all the information as itβs written here. However, we could rewrite the
equation so that cos of π΄ is by itself.

If we subtract π squared and π
squared from both sides of the equation, we end up with π squared minus π squared
minus π squared equals negative two ππ times cos of π΄. So we divide both sides of the
equation by negative two ππ. Negative two ππ divided by
negative two ππ just leaves cos of π΄. And so cos of π΄ is equal to π
squared minus π squared minus π squared over negative two ππ, substituting in
the values we know for π, π, and π and then using a calculator to calculate
this. To do that accurately, make sure
you put some form of brackets or parentheses around your numerator and your
denominator. When we do that, we get the cos of
π΄ equals 0.1228042 continuing.

Since weβre interested in the
measure of this angle, we need to take the inverse cosine of both sides of this
equation. This tells us that the measure of
angle π΄ is 82.9460 continuing degrees. We need to round this to the
nearest degree. So weβll say that π΄ equals 83
degrees. And then weβll add that value to
our diagram. Of course, we could use the law of
cosines a third time to find the measure of angle π΅. However, because weβre working
inside a triangle, we know that the three angles must sum together to be 180
degrees. And so we can solve for the measure
of angle π΅ by subtracting 83 degrees and 60 degrees from 180 degrees. When we do that, we find the
measure of angle π΅ is equal to 37 degrees.

And so we found our three unknown
values for this triangle. Side length π equals 24.434
centimetres, measure of angle π΄ is 83 degrees, and the measure of angle π΅ is 37
degrees.