The question is find the general solution of this equation: \sin(7\phi)+\cos(3\phi)=0

Umaiza Hutton

Umaiza Hutton

Answered question

2022-02-26

The question is find the general solution of this equation:
sin(7ϕ)+cos(3ϕ)=0

Answer & Explanation

litoshypinaylh4

litoshypinaylh4

Beginner2022-02-27Added 6 answers

cos7ϕ+sin3ϕ=0
Note that 7ϕ=5ϕ+2ϕ and that 3ϕ=5ϕ2ϕ
Rewrite original statement as: cos(5ϕ+2ϕ)+sin(5ϕ2ϕ)=0
cos5ϕcos2ϕsin5ϕsin2ϕ+sin5ϕcos2ϕsin5ϕsin2ϕ=0
cos5ϕ(cos2ϕsin2ϕ)+sin5ϕ(cos2ϕsin2ϕ)=0
(cos5ϕ+sin5ϕ)(cos2ϕsin2ϕ)=0
Either cos5ϕ+sin5ϕ=0
sin5ϕ=cos5ϕ
tan5ϕ=1
5ϕ=π4+kπ
ϕ=π20+kπ5
or cos2ϕsin2ϕ=0
sin2ϕ=cos2ϕ
tan2ϕ=1
2ϕ=π4+kπ
ϕ=π8+kπ2

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