A triangle has sides A, B, and C. Sides A and B are of lengths #7# and #5#, respectively, and the angle between A and B is #(5pi)/8 #. What is the length of side C?

Well this is a very tricky one, i wish i can draw the Triangle to explain better, but with the theory i hope you will understand..

Firstly Note that the Angle is in Radians..

If you observe closely,

Angel between A and B is #color(white)(x)5pi/8#

You're aware that there are only 3 angels which are #A, B, C#

Now again; What is the Angle between #A and B# ?

Simple, the answer is #C#

Cause it's a Triangle so the angle is C, since it's in between the two angles..

It's just a trick to push you off course from the desired Answer..

Hence we have as follows;

#C^o = 5pi/8#

#a = 7#

#b = 5#

#c = ?#

Before we proceed further, remember first i said Note that the Angle is in Radians..

We must first convert from Radians to Degree..

Using this formula

#Degree rArr radians xx 180/pi#

Hence #rArr# #C^o = (5pi)/8 xx 180/pi#

#C^o = (5cancelpi)/8 xx 180/cancelpi#

#C^o = 5/8 xx 180/1#

#C^o = (5 xx 180)/8#

#C^o = 900/8#

#C^o = 900/8#

#C^o = 112.5#

#C^o ~~ 113#

Now to find c we shall using the Cosine Rule Formula

#c^2 = a^2 + b^2 - 2ab Cos(C)#

Imputing your values..

#c^2 = 7^2 + 5^2 - 2 xx 7 xx 5 Cos(113)#

#c^2 = 49 + 25 - 70 (-0.39073)#

#c^2 = 74 + 27.3511#

#c^2 = 101.3511#

#c^2 ~~ 101#

#c = sqrt101#

#c = 10.04987#

#c ~~ 10#

#:. c = 10 -> Answer#