Video Transcript
For the triangle given, find the
value of the three angles to the nearest degree.
In the figure, we are given the
lengths of all three sides of our triangle. Side length 𝐴 opposite angle 𝐴 is
equal to 27 centimeters. Side length 𝐵 is equal to 28
centimeters. And side length 𝐶 is equal to nine
centimeters. We need to calculate the measure of
angle 𝐴, angle 𝐵, and angle 𝐶. We can do this using the law of
cosines, which states that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus
two 𝑏𝑐 multiplied by the cos of angle 𝐴. The lowercase letters here
correspond to the side lengths and the capital or uppercase 𝐴 is an angle.
When we wish to calculate the
measure of an angle as opposed to the length of a side, we can use a rearrangement
of this formula. This states that the cos of angle
𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all divided by two
𝑏𝑐. We will begin by substituting our
values from the figure to calculate the measure of angle 𝐴. The cos of angle 𝐴 is equal to 28
squared plus nine squared minus 27 squared all divided by two multiplied by 28
multiplied by nine. Typing the right-hand side into our
calculator gives us 17 over 63. We can then take the inverse cos of
both sides of this equation such that angle 𝐴 is equal to the inverse cos of 17
over 63. Typing this into the calculator
gives us 74.3451 and so on.
We are asked to give the value of
the three angles to the nearest degree. We can therefore conclude that the
measure of angle 𝐴 to the nearest degree is 74 degrees. We can now repeat this process to
calculate the measure of angle 𝐵. The cos of angle 𝐵 is equal to 𝑎
squared plus 𝑐 squared minus 𝑏 squared all divided by two 𝑎𝑐. In this question, we have 27
squared plus nine squared minus 28 squared all divided by two multiplied by 27
multiplied by nine. This time, the right-hand side
simplifies to 13 over 243. Angle 𝐵 is equal to the inverse
cos of 13 over 243. Typing this into the calculator
gives us 86.9333 and so on. Once again, we need to round to the
nearest degree such that the measure of angle 𝐵 is 87 degrees.
Finally, we can repeat this process
to calculate the measure of angle 𝐶. The cos of angle 𝐶 is equal to 27
squared plus 28 squared minus nine squared all divided by two multiplied by 27
multiplied by 28. The right-hand side of this
equation is equal to 179 over 189. Therefore, angle 𝐶 is equal to the
inverse cos of 179 over 189. Rounding our value of 18.7214 and
so on to the nearest degree, we see that the measure of angle 𝐶 is 19 degrees.
At this stage, it is worth
recalling that the three angles in a triangle must sum to 180 degrees. Whilst we need to use the exact
values to check this, adding 74, 87, and 19 will give us a good indication as to
whether our answers are correct. In this case, the three angles do
sum to 180 degrees. In general, provided we are close
to 180, our answers are likely to be correct. In this question, the measure of
angle 𝐴 is 74 degrees, angle 𝐵 is 87 degrees, and angle 𝐶 is 19 degrees, where
all three are measured to the nearest degree.
