In triangle ABC we have angle C = 3 times angle A, a=27 and c=48 What is b? Note: a is the side length opposite A etc. please help

  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

__

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)

__

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.