`sin(2x)sin(x) = cos(x)` Find the exact solutions of the equation in the interval [0, 2pi).

Tuesday, August 4, 2015

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

$\displaystyle{\sin{{\left({2}{x}\right)}}}{\sin{{\left({x}\right)}}}={\cos{{\left({x}\right)}}},{0}\le{x}\le{2}\pi$

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

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

$\displaystyle={\cos{{\left({x}\right)}}}{\left({2}{{\sin}}^{{2}}{\left({x}\right)}-{1}\right)}={0}$

$\displaystyle={\cos{{\left({x}\right)}}}{\left(\sqrt{{{2}}}{\sin{{\left({x}\right)}}}-{1}\right)}{\left(\sqrt{{{2}}}{\sin{{\left({x}\right)}}}+{1}\right)}={0}$

solving each part separately,

$\displaystyle{\cos{{\left({x}\right)}}}={0},{\left(\sqrt{{{2}}}{\sin{{\left({x}\right)}}}-{1}\right)}={0},{\left(\sqrt{{{2}}}{\sin{{\left({x}\right)}}}+{1}\right)}={0}$

General solutions for cos(x)=0 are,

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

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

$\displaystyle{x}=\frac{\pi}{{2}},{x}={\left({3}\frac{\pi}{{2}}\right)}$

$\displaystyle{\left(\sqrt{{{2}}}{\sin{{\left({x}\right)}}}-{1}\right)}={0}$

$\displaystyle{\sin{{\left({x}\right)}}}=\frac{{1}}{\sqrt{{{2}}}}$

General solutions are,

$\displaystyle{x}=\frac{\pi}{{4}}+{2}\pi{n},{x}=\frac{{{3}\pi}}{{4}}+{2}\pi{n}$

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

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

$\displaystyle\sqrt{{{2}}}{\sin{{\left({x}\right)}}}+{1}={0}$

$\displaystyle{\sin{{\left({x}\right)}}}=-\frac{{1}}{\sqrt{{{2}}}}$

General solutions are,

$\displaystyle{x}=\frac{{{5}\pi}}{{4}}+{2}\pi{n},{x}=\frac{{{7}\pi}}{{4}}+{2}\pi{n}$

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

$\displaystyle{x}=\frac{{{5}\pi}}{{4}},{x}=\frac{{{7}\pi}}{{4}}$

Combine all the solutions,

$\displaystyle{x}=\frac{\pi}{{2}},{x}=\frac{{{3}\pi}}{{2}},{x}=\frac{\pi}{{4}},{x}=\frac{{{3}\pi}}{{4}},{x}={\left({5}\frac{\pi}{{4}}\right)},{x}=\frac{{{7}\pi}}{{4}}$

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Tuesday, August 4, 2015

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

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