How do you find the angles of a triangle given sides #a=8, b=10, c=12#?

1 Answer
Aug 27, 2016

Answer:

#hat A = 41.41^@ , hat B = 55.77^@, hat C = 82.82^@#

Explanation:

With the help of sinus law that establishes

#a/sin(hat A) = b/sin(hat B) = c/sin(hat C)#

and also

#hatA + hatB + hatC = pi#

The formulation is

#{ (a/sin(hat A) = b/sin(hat B)), (b/sin(hat B) = c/sin(hat C)), (hatA + hatB + hatC = pi) :}#

By knowing that

#sin(pi-(hat A+ hat B))=sin(hat A+ hat B)#

This system can be reduced to

#{ (a/sin(hat A) = b/sin(hat B)), (b/sin(hat B) = c/sin(hat A + hat B)) :}#

Solving for #hat A, hat B# after substituting

#sin(hat A+ hat B) = sin(hat A)cos(hat B)+sin(hat B)cos(hat A)#
and #cos(alpha) = sqrt(1-sin(alpha)^2)# we have:

#hat A = arctan((b^2+c^2-a^2)/(bc), sqrt((b+c-a)(a+b-c)(a-b+c)(a+b+c))/(bc))#
#hat B = arctan((a^2-b^2+c^2)/(ac),sqrt((b+c-a)(a+b-c)(a-b+c)(a+b+c))/(ac)#

or substituting values

#hat A = arctan(sqrt(7)/3) =41.41^@ , hat B = arctan(5 sqrt(7)/9)=55.77^@# and #hat C = 180^@-(41.41^@ +55.77^@) = 82.82^@#