Solve each equation, giving all solutions (general solutions), tan^5(3x)=9tan(3x)

slaggingV 2021-02-08 Answered
Solve each equation, giving all solutions (general solutions), tan5(3x)=9tan(3x)
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Expert Answer

nitruraviX
Answered 2021-02-09 Author has 101 answers

Let's make it simple and let y=tan(3x), then y5=9y, so y(y49)=y(y23)(y2+3)=0.
So y=0,±3 are the real roots.
Now we go back to y=tan(3x)=0,±3.
So x=0,tan(3x)=±3.tan(60)=3, so 3x=60andx=20 degrees. But we know that tangent is negative in QII and QIV. So 3x=18060=120andx=40 degrees, 3x=36060=300 so x=100 degrees.
CHECK: x=0: tan(0)=0 so tan5(3x)=tan(3x)=0.
x=20: 3x=60, tan(60)=3andtan5(60)=93=9tan(60).
x=40: 3x=120, tan(120)=3andtan5(120)=93=9tan(120).
x=100: 3x=300, tan(300)=3andtan5(300)=93=9tan(300).
But we can add 360 to 0, 60, 120 and 300, then divide by 3 to find all possible angles, so the solution is:
x=(13)(360n+0)=120n,(13)(360n+60)=120n+20,120n+40,120n+100 where n is an integer.

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