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

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

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

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

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

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

$B=4/sqrt(2)$

$B=2sqrt(2)$

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