Joslyn Landry

2023-03-25

The solution set of $\mathrm{tan}\theta =3\mathrm{cot}\theta $ is

Eden Pacheco

Beginner2023-03-26Added 4 answers

The equation becomes: using the reciprocal identity $\mathrm{cot}=\frac{1}{\mathrm{tan}}$

tanθ - 3/tanθ = 0

Then, multiplying both sides by tanθ (assuming that $\mathrm{tan}\theta \ne 0$ doesn't solve the resulting equation, which it won't) yields:

${\mathrm{tan}}^{2}-3=0$.

Solving this for $\mathrm{tan}\theta $ yields $\mathrm{tan}\theta =\pm \surd 3$.

From the unit circle, if θ is in [0, 2π], tanθ = -√3 solves to yield θ = 2π/3 and θ = 5π/3 and tanθ = √3 solves to yield θ = π/3 and θ = 4π/3. If you want the general solution, you can get this by adding any multiple of π to these solutions.

tanθ - 3/tanθ = 0

Then, multiplying both sides by tanθ (assuming that $\mathrm{tan}\theta \ne 0$ doesn't solve the resulting equation, which it won't) yields:

${\mathrm{tan}}^{2}-3=0$.

Solving this for $\mathrm{tan}\theta $ yields $\mathrm{tan}\theta =\pm \surd 3$.

From the unit circle, if θ is in [0, 2π], tanθ = -√3 solves to yield θ = 2π/3 and θ = 5π/3 and tanθ = √3 solves to yield θ = π/3 and θ = 4π/3. If you want the general solution, you can get this by adding any multiple of π to these solutions.

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