Question

The lengths of the sides of a triangle are 7, 8 and 9 cm. How do you calculate the size of the smallest angle in the triangle?

Answer 1

Use the Law of Cosines (and a calculator) to determine:
smallest angle `~~0.841069` radians

Explanation:

Denote the sides as
`color(white)("XXX")a=9`
`color(white)("XXX")b=8`
`color(white)("XXX")c=7`
and
`color(white)("XXX")`the angle opposite `a` as `A`
`color(white)("XXX")`the angle opposite `b` as `B`
`color(white)("XXX")`the angle opposite `c` as `C`

Note that the smallest angle is the angle opposite the shortest side (i.e. angle `C` in this case).

By The Law of Cosines:
`color(white)("XXX")c^2=a^2+b^2-2ab*cos(C)`
or
`color(white)("XXX")cos(C)=((a^2+b^2)-c^2)/2ab`

Using the given values
`color(white)("XXX")cos(C)=(8^2+9^2-7^2)/(2*8*9)=96/144=2/3`

If `cos(C)=2/3` then
`color(white)("XXX")C=arccos(2/3)` (here is where I used a calculator)
`color(white)("XXXX")~~0.841069` radians