Finding solution of a Trigonometric equation \tan A+\tan 2A+\tan 3A=0 I tried

Gregory Jones 2021-12-28 Answered
Finding solution of a Trigonometric equation
tanA+tan2A+tan3A=0
I tried converting these all in sin and cos and I got the answer but the answer didn't match
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Expert Answer

Hattie Schaeffer
Answered 2021-12-29 Author has 37 answers
Use this:
tan3A=tanA+tan2A1tanAtan2A
Substitute the numerator with −tan3A and youre
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enhebrevz
Answered 2021-12-30 Author has 25 answers

tanA+tan2A=sin3AcosAcos2A
tanA+tan2A+tan3A=0sin3AcosAcos2A+sin3Acos3A=0
sin3A(cos3A+cosAcos2A)=0
If sin3A=0,  3A=nπ where n is any integer
Otherwise,
cos3A+cosAcos2A=04cos3A3cosA+cosA(2cos2A1)=0
cosA(6cos2A4)=0
If cosA=0,   A=(2m+1)π2
Otherwise, 6cos2A4=0cos2A=23,  cos2A=2cos2A1=13

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nick1337
Answered 2022-01-08 Author has 510 answers

Hint
The following identities for the tangent of multiple angles should be useful
tan2A=2tanA12tan2A
tan3A=3tanAtan3A13tan2A
I am sure that you can take from here.

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