Question

How do you solve the triangle when given a=5, b=7, and c=6?

Answer 1

Use the Law of Cosines to determine the angles

Explanation:

Law of Cosines
`color(white)("XXX")c^2=a^2+b^2-2abcos(/_C)`
`rArr`
`color(white)("XXX")cos(C) = (a^2+b^2-c^2)/(2ab)`

Similarly
`color(white)("XXX")cos(A) = (b^2+c^2-a^2)/(2bc)`
and
`color(white)("XXX")cos(B)= (a^2+c^2-b^2)/(2ac)`

Substituting the given values for `a, b, and c`
we get
`color(white)("XXX")cos(/_A) = 0.714`
then taking the `arccos` of both sides:
`color(white)("XXX")/_A = arcos(0.715) = 0.775 " radians" = 44.4^@`

Similarly we can find:
`color(white)("XXX")/_B = 1.369 " radians" = 78.5^@`
and
`color(white)("XXX")/_C = 0.997 " radians" = 57.1^@`