A triangle has sides A, B, and C. The angle between sides A and B is #pi/3#. If side C has a length of #8 # and the angle between sides B and C is #pi/12#, what is the length of side A?

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Answer 1 · 2017-05-18T17:30:55

#Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)color(red)(2.391)#

#Bcolor(white)(00)color(black)((7pi)/12)color(white)(00000)bcolor(white)(00)8.923#

#Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8#

Let's write this information in a table, where capital letters correspond to angles , lowercase to lengths :

#Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)#

#Bcolor(white)(00)color(white)(pi/12)color(white)(00000)bcolor(white)(00)#

#Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8#

All angles in a triangle add up to #180^o#, or #pi#

#pi/12+pi/3# only equals #(5pi)/12#.

#(12pi)/12-(5pi)/12=(7pi)/12#

That's the remaining angle:

#Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)#

#Bcolor(white)(00)color(black)((7pi)/12)color(white)(00000)bcolor(white)(00)#

#Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8#

We can use law of sines to find the other lengths:

#(sin(pi/3))/8=(sin(pi/12))/a#

#a~~2.391#

Just, for fun, let's also find length b

#(sin(pi/3))/8=(sin((7pi)/12))/b#

#b~~8.923#