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a)  Use the Law of Cosines:  c²  =  a² + b² - 2·a·b·cos(C)

 substitue in the values:         9²  =  5² + 8² - 2·5·8·cos(C)

 evaluate:                              81  =  25 + 64 - 80·cos(C)

 add:                                      81  =  89 - 80·cos(C)

 subtract:                                -8  =  -80·cos(C)

 divide:                                1/10  =  cos(C)

                                  cos^-1(1/10)  =  C

                                                C  =  84.26^o 

 

You can now find either angle(B) or angle(C) by using the Law of Sines or the Law of Cosines.

Then, you can find the remaining angle by subtaction from 180^o.

Hint:  if you are to find all three angles, find the largest angle first by the Law of Cosines; using the Law of Sines to find the largest angle can give you an incorrect answer if the angle is greater than 90^o.

 

b)  Use the Law of Sines:  sin(B) / 5  =  sin(40^o) / 4

cross multiply:                     4·sin(B)  =  5·sin(40^o)

divide:                                     sin(B)  =  5·sin(40^o) / 4

                                                      B  =  sin^-1( 5·sin(40^o) / 4 )

                                                      B  =  53.46^o

 

Now, find the third angle by subtaction from 180^o.

Then, find the last side by the Law of Sines.