Trig solve these triangles any or all

  see below

Step-by-step explanation:

21) The law of sines can be used, since you have a side and its opposite angle.

  sin(F)/DE = sin(D)/EF

  F = arcsin(DE/EF·sin(D)) = arcsin(20/31·sin(95°)) ≈ 39.994°

  E = 180° -95° -39.994° ≈ 45.006°

  DF = sin(45.006°)/sin(95°)·31 ≈ 22.006

__

22) The remaining two problems can be solved using the law of cosines:

  c^2 = a^2 + b^2 - 2ab·cos(C)

Of course, c is the square root of the expression on the right.

  EF = √(19^2 +35^2 -2(19)(35)cos(61°)) ≈ √(941.203) ≈ 30.679

Then an angle can be found using the law of sines

  E ≈ arcsin(35/30.679·sin(61°)) ≈ 86.203°

  F ≈ 180° -61° -86.203° ≈ 32.797°

__

23) As in 22 …

  RS = √(20^2 +28^2 -2(20)(28)cos(91°)) ≈ √(1203.547) ≈ 34.692

  R ≈ arcsin(20/34.692·sin(91°)) ≈ 35.199°

  S ≈ 180° -91° -35.199° ≈ 53.801°

2^y = ln 5, or   y(ln 2) = ln 5.   solving for y,   y = (ln 5) / (ln 2).   y=5

step-by-step explanation: