A triangle has sides A, B, and C. The angle between sides A and B is #(3pi)/4#. If side C has a length of #4 # and the angle between sides B and C is #pi/12#, what are the lengths of sides A and B?

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Answer 1 · 2016-07-21T06:20:54

#A=2sqrt(3)-2#; #B=2sqrt(2)#

Let #hat(AB)=(3pi)/4; C=4; hat(BC)=pi/12#

Then you can use the theorem of Euler:

#a/sinalpha=b/sinbeta=c/singamma#

and you will have

#A/sin hat(BC)=C/sin hat(AB)#

to find A.

Let's substitute the known values

#A/sin(pi/12)=4/sin((3pi)/4)#

#A=4sin(pi/12)/sin((3pi)/4)#

#A=cancel4((sqrt(6)-sqrt(2))/cancel4)/(sqrt(2)/2)#

#A=2(sqrt(6)-sqrt(2))/sqrt(2)#

#A=sqrt(2)(sqrt(6)-sqrt(2))#

#A=2sqrt(3)-2#

To find B, you can find the opposite angle #hat(AC)# and use the same processing

#hat(AC)=pi-(pi/12+(3pi)/4)=pi/6#

#B/sin hat(AC)=C/sin hat(AB)#

#B=(Csin hat(AC))/sin hat(AB)#

#B=(4sin (pi/6))/sin ((3pi)/4)#

#B=(4*1/2)/(sqrt(2)/2)#

#B=4/sqrt(2)#

#B=2sqrt(2)#