In triangle ABC, if the length of side #b# is 3 centimeters and the measures of #/_B# and #/_C# are 45° and 60°, respectively, what is the length of side #c# to two decimal places?

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Answer 1 · 2016-10-18T02:53:37

#3.67# cm

We need to use the law of sines, #a/SinA# = #b/SinB# = #c/SinC#

Substituting for #hatB and hatC#, we get #3/Sin45 = c/Sin60#

Now convert the special angles: #3/(sqrt2/2)# = #c/(sqrt3/2)#

Multiply both sides by #sqrt3/2# to get #c# by itself and simplify:

#c=(sqrt3/2) xx 3/(sqrt2/2)#

#c = (3sqrt3)/2/(sqrt2/2)# Cancel out the denominators

=#(3sqrt3)/(sqrt2)# Multiply top and bottom by #(sqrt2)#

=#(3sqrt6)/2#. Now we can check our answer

#3/(sqrt2/2)# = #(3sqrt6/2)/(sqrt3/2)#

#sqrt6/sqrt3=sqrt2#

#3/(sqrt2/2)# = #3/(sqrt2/2)# ✓

So, #c=(3sqrt6)/2≈ 3.67# cm