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?

1 Answer
Alan P.
Jun 23, 2016

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

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