General Solution of Equation tanθ = tanα

General Solution of Equation tanθ = tanα

General Solution of Equation: Given tanθ = tanα

\(\frac{\sin \theta }{\cos \theta }=\frac{\sin \alpha }{\cos \alpha }\),

sin θ cosα = sinα cosθ,

sin θ cosα – sinα cosθ = 0,

\(\left( \because \sin \left( A-B \right)=\sin A\cos B-\cos A\sin B \right)\),

sin (θ – α) = 0 ,

θ – α = Sin⁻¹(0),

\(\theta -\alpha =n\pi ,\ n\in Z\),

\(\theta =n\pi +\alpha ,\ n\in Z\),

Note: For general solution of the equation tanθ = k, where k ϵ R, We have tanθ = tan(tan⁻¹k). thus, \(\theta =n\pi +({{\tan }^{-1}}k),\ n\in Z\),

Example: Solve tan5θ = cot2θ

Solution:  tan5θ = cot2θ

\(\tan 5\theta =\tan \left( \frac{\pi }{2}-2\theta  \right)\),

\(5\theta =n\pi +\left( \frac{\pi}{2}-2\theta  \right)\),

7θ =nπ +  π/2,

θ =nπ/7 +  π/14, where n ϵ Z, but n ≠ 3, 10, 17, . ……

where tan5θ is not defined.