Video Transcript
Find the cos of 𝐴 plus 𝐵 given
the cos of 𝐴 equals 15 over 17 and the cos of 𝐵 equals five over 13, where 𝐴 and
𝐵 are acute angles.
When we see this cos of 𝐴 plus 𝐵,
it should remind us of our angle sum identities. We know that the cos of 𝐴 plus 𝐵
is equal to the cos of 𝐴 times the cos of 𝐵 minus the sin of 𝐴 times the sin of
𝐵. We have enough information for the
first term since we’ve already been given the value of cos of 𝐴 and cos of 𝐵. So how should we go about finding
the sin of 𝐴 and the sin of 𝐵 if we know the cosine ratios? They are acute angles. And therefore, one strategy to find
sin 𝐴 and sin 𝐵 would be to make right-angled triangles with these
proportions.
First, we can draw a right triangle
with angle 𝐴. We know that the cosine
relationship will be equal to the adjacent side length over the hypotenuse. And so we would label the adjacent
side to 𝐴 15 and the hypotenuse 17. To find the sine relationship,
we’ll need to know this opposite side length, and that means we’ll need to use the
Pythagorean theorem. We’ll let our unknown side be
lowercase 𝑎. And then we’ll have 17 squared
equals 𝑎 squared plus 15 squared, which will give us 289 is equal to 𝑎 squared
plus 225. To isolate 𝑎, we subtract 225 from
both sides, and then we get 64 is equal to 𝑎 squared. Taking the square root of both
sides, we get 𝑎 equal to eight.
We’re only interested in the
positive square root since we’re dealing with distance. If 𝑎 equals eight, then the sin of
angle 𝐴 will be equal to eight over 17. If we consider a second
right-angled triangle with angle 𝐵, its adjacent side length is five and its
hypotenuse is 13. We should recognize that this is a
Pythagorean triple. It’s a set of positive integers
that occur in the ratio 𝑎 squared plus 𝑏 squared equals 𝑐 squared. And because we know the hypotenuse
is 13 and one of the sides is five, this is a five, 12, 13 triangle. And therefore, the adjacent side
length will be 12 and the sin of angle 𝐵 is 12 over 13. And we can plug that value in. 15 over 17 times five over 13 is 75
over 221. Eight over 17 times 12 over 13 is
96 over 221, which makes the cos of 𝐴 plus 𝐵 equal to negative 21 over 221.
