Answer:
35
Step-by-step explanation:
The law of sines tells us ...
sin(C)/c = sin(A)/a
a·sin(3A) = c·sin(A)
Using the identity sin(3x) = 3cos(x)·sin(x) -sin(x)^3 and sin(x)^2 +cos(x)^2 = 1, we can simplify this to ...
sin(A)(4cos(A)^2 -1) = (c/a)sin(A)
4cos(A)^2 = c/a +1 = (48+27)/27 = 75/27 = 25/9
cos(A)^2 = 25/36
cos(A) = 5/6
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Now, the angle B will be the difference between 180° and the sum of the other two angles:
B = 180° -A -3A = 180° -4A
Using appropriate trig identities, we can write ...
sin(B) = 4cos(A)^3sin(A) -4sin(A)^3cos(A)
= 4sin(A)cos(A)(cos(A)^2 -sin(A)^2)
= 4sin(A)cos(A)(2cos(A)^2 -1)
Filling in our value for cos(A), this becomes ...
sin(B) = 4sin(A)(5/6)(2(5/6)^2-1) = sin(A)(35/27)
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The law of sines tells us ...
b/sin(B) = a/sin(A)
b = a·sin(B)/sin(A) = 27(35/27)sin(A)/sin(A) = 35
The length of side b is 35 units.
