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ElskerNorge
@ElskerNorge
August 2022
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(sinx+sin3x+sin5x)/(cosx+cos3x+cos5x) + 2tgx=0
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sedinalana
Verified answer
(sinx+sin3x+sin5x)/(cosx+cos3x+cos5x) + 2tgx=0
(2sin3xcos2c+sin3x)/(2cos3xcos2x+cos3x)+2tgx=0
sin3x(2cos2x+1)/cos3x(2cos2x+1)+2tgx=0
tg3x+2tgx=0
(3tgx-tg
³x)/(1-3tg²x)+2tgx=0
1-3tg²x≠0⇒tgx≠+-1/√3⇒x≠+-π/6+πk,k∈z
(3tgx-tg³x)+2tgx(1-3tg²x)=0
3tgx-tg³x+2tgx-6tg³x=0
5-7tg²x=0
7tg²x=5
tg²x=5/7
tgx=-√35/7⇒x=-arctg√35/7+πk,k∈z
tgx=√35/7⇒x=arctg√35/7+πk,k∈z
Ответ x={πk;-arctg√35/7+πk;arctg√35/7+πk,k∈z}
2 votes
Thanks 3
Denik777
Надо упомянуть, что еще должно выполняться cos(2x)≠-1/2.
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Answers & Comments
Verified answer
(sinx+sin3x+sin5x)/(cosx+cos3x+cos5x) + 2tgx=0(2sin3xcos2c+sin3x)/(2cos3xcos2x+cos3x)+2tgx=0
sin3x(2cos2x+1)/cos3x(2cos2x+1)+2tgx=0
tg3x+2tgx=0
(3tgx-tg³x)/(1-3tg²x)+2tgx=0
1-3tg²x≠0⇒tgx≠+-1/√3⇒x≠+-π/6+πk,k∈z
(3tgx-tg³x)+2tgx(1-3tg²x)=0
3tgx-tg³x+2tgx-6tg³x=0
5-7tg²x=0
7tg²x=5
tg²x=5/7
tgx=-√35/7⇒x=-arctg√35/7+πk,k∈z
tgx=√35/7⇒x=arctg√35/7+πk,k∈z
Ответ x={πk;-arctg√35/7+πk;arctg√35/7+πk,k∈z}