Solve 2+\cot\theta=\csc\theta where 0\leq\theta<2\pi ?

manhangasto6

manhangasto6

Answered question

2022-02-26

Solve
2+cotθ=cscθ
where
0θ<2π ?

Answer & Explanation

Nathan Kent

Nathan Kent

Beginner2022-02-27Added 8 answers

We know csc2θcot2θ=1
(cscθ+cotθ)(cscθcotθ)=1
Given cscθcotθ=2
So, cscθ+cotθ=12
So, 2cscθ=52sinθ=45>0
cotθ=12cscθ=1254<0
So, θ will lie in the 2nd quadrant, the principal value in (π2,π)
Elodie Williamson

Elodie Williamson

Beginner2022-02-28Added 7 answers

2+cotθ=cscθ2+cosθsinθ=1sinθ
2sinθ+cosθ=1(sinθ1)
25sinθ+15cosθ=15
cosasinθ+sinacosθ=15 where a=arctan(12)
sin(a+θ)=15
I think you can solve it from here.
Alternatively,
Let cotθ=x, then, csc2θ=1+cot2θ=1+x2
Thus, (2+cotθ)2=csc2θ
4+x2+4x=x2+1
4x=3x=34tanθ=43
θ=πarctan(43), 2πarctan(43)
But only πarctan(43) satisfies the original equation.Thus, the solution is θ=πarctan(43)

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