Find all solutions of the equation in interval 02pi Write yo

Find all solutions of the equation in interval [0,2pi). Write your answer as a comma-separated list.

Solution

[0,2)

1)3tan³x =tanx

3tan³x -tanx=0

tanx(3tan²x-1)=0

tanx =0 , tan²x=1/3

tanx =0 , tanx =-1/3, tanx =1/3

x =0,/6,5/6,,7/6,11/6

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2)4cos²x+2cosx-2=0

2cos²x+cosx-1=0

2cos²x+2cosx-cosx-1=0

2cosx(cosx+1)-1(cosx+1)=0

(cosx+1)(2cosx-1)=0

cosx=-1, cosx=1/2

x= , x=/3,5/3

x=/3 , ,5/3

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3)10sec²x+5tan²x-15=0

=>10(1+tan²x)+5tan²x-15=0

=>10+10tan²x+5tan²x-15=0

=>15tan²x-5=0

=>tan²x=5/15

=>tan²x=1/3

=>tanx =-1/3 ,tanx =1/3

=>x =2/3 ,5/3 , x=/3,43

=>x=/3,2/3 ,4/3 ,5/3