`tan(2x) - cot(x) = 0` Find the exact solutions of the equation in the interval [0, 2pi).

Tuesday, April 29, 2008

$\displaystyle{\tan{{\left({2}{x}\right)}}}-{\cot{{\left({x}\right)}}}={0}$ Find the exact solutions of the equation in the interval [0, 2pi).

$\displaystyle{\tan{{\left({2}{x}\right)}}}-{\cot{{\left({x}\right)}}}={0}$

express in terms of sin and cos,

$\displaystyle\frac{{\sin{{\left({2}{x}\right)}}}}{{\cos{{\left({2}{x}\right)}}}}-\frac{{\cos{{\left({x}\right)}}}}{{\sin{{\left({x}\right)}}}}={0}$

$\displaystyle\frac{{{\sin{{\left({x}\right)}}}{\sin{{\left({2}{x}\right)}}}-{\cos{{\left({x}\right)}}}{\cos{{\left({2}{x}\right)}}}}}{{{\cos{{\left({2}{x}\right)}}}{\sin{{\left({x}\right)}}}}}={0}$

$\displaystyle{\sin{{\left({x}\right)}}}{\sin{{\left({2}{x}\right)}}}-{\cos{{\left({x}\right)}}}{\cos{{\left({2}{x}\right)}}}={0}$

$\displaystyle-{\left({\cos{{\left({x}\right)}}}{\cos{{\left({2}{x}\right)}}}-{\sin{{\left({x}\right)}}}{\sin{{\left({2}{x}\right)}}}\right)}={0}$

using the identity $\displaystyle{\cos{{A}}}{\cos{{B}}}-{\sin{{A}}}{\sin{{B}}}={\cos{{\left({A}+{B}\right)}}}$

$\displaystyle\Rightarrow-{\cos{{\left({x}+{2}{x}\right)}}}={0}$

$\displaystyle\Rightarrow{\cos{{\left({3}{x}\right)}}}={0}$

General solutions for cos(3x)=0 are,

$\displaystyle{3}{x}=\frac{\pi}{{2}}+{2}\pi{n},{x}=\frac{{{3}\pi}}{{2}}+{2}\pi{n}$

$\displaystyle{x}=\frac{{{4}\pi{n}+\pi}}{{6}},{x}=\frac{{{4}\pi{n}+{3}\pi}}{{6}}$

Solutions for the range $\displaystyle{0}\le{x}\le{2}\pi$ are,

$\displaystyle{x}=\frac{\pi}{{6}},\frac{\pi}{{2}},\frac{{{5}\pi}}{{6}},\frac{{{7}\pi}}{{6}},\frac{{{3}\pi}}{{2}},\frac{{{11}\pi}}{{6}}$

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Tuesday, April 29, 2008

$\displaystyle{\tan{{\left({2}{x}\right)}}}-{\cot{{\left({x}\right)}}}={0}$ Find the exact solutions of the equation in the interval [0, 2pi).

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