Question

How do you find the measure of each of the angles of a triangle given the measurements of the sides are 12, 20, 22?

Answer 1

Use the Law of Cosines as demonstrated below.

Explanation:

Let `/_A` be the angle opposite side `abs(a)=22`
and `/_B` be the angle opposite side `abs(b)=20`
and `/_C` be the angle opposite side `abs(c)=12`

The Law of Cosines tells us that
`color(white)("XXX")abs(a)^2=abs(b)^2+abs(c)^2-2abs(b)abs(c)cos(A)`
or (in a form more useful for this problem):
`color(white)("XXX")cos(/_A)= (abs(b)^2+abs(c)^2-abs(a)^2)/(2abs(b)abs(c))`
or...
`color(white)("XXX")/_A = "arccos"((abs(b)^2+abs(c)^2-abs(a)^2)/(2abs(b)abs(c)))`

Using the given values (and a calculator)
`color(white)("XXX")/_A ~~ 1.445468` (radians)

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Similarly, we can find:
`color(white)("XXX")/_B~~1.124289` (radians)
and
`color(white)("XXX")/_C~~0.571835` (radians)