How to solve \(\displaystyle{\frac{{{1}}}{{{\sin{{x}}}}}}+{\frac{{\sqrt{{{3}}}}}{{{\cos{{x}}}}}}={4}\)

mwombenizhjb

mwombenizhjb

Answered question

2022-03-31

How to solve
1sinx+3cosx=4

Answer & Explanation

Adan Berry

Adan Berry

Beginner2022-04-01Added 12 answers

We may assume x≠∈π2Z, hence your equation is equivalent to:
cos(x)+3sin(x)=4sin(x)cos(x)=2sin(2x)
or to:
sin(x+π6)=sin(2x)
whose fundamental solutions are given by 2x=x+π6, leading to x=π6, and by
x+π6=(2k+1)π2x, leading to x[5π18,17π18,29π18]. So we get:
x[π6,5π18,17π18,29π18]+2πZ

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