Notes: `cos(x + pi/4) - cos(x - pi/4) = 1` Find all solutions of the equation in the interval [0, 2pi).

Tuesday, 24 February 2015

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1}$ Find all solutions of the equation in the interval [0, 2pi).

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1},{0}\le{x}\le{2}\pi$

We will use the following identity,

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

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1}$

$\displaystyle\Rightarrow{\left({\cos{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left({x}\right)}}}-{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\sin{{\left({x}\right)}}}\right)}-{\left({\cos{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left({x}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\sin{{\left({x}\right)}}}\right)}={1}$

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

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

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

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

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,

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

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1},{0}\le{x}\le{2}\pi$

We will use the following identity,

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

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1}$

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

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

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

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

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,

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

Notes

Tuesday, 24 February 2015

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1}$ Find all solutions of the equation in the interval [0, 2pi).

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Is there any personification in "The Tell-Tale Heart"?

Tuesday, 24 February 2015

Find all solutions of the equation in the interval [0, 2pi).

No comments:

Post a Comment

Is there any personification in &quot;The Tell-Tale Heart&quot;?

$\displaystyle{\cos{{\left({x}+\frac{\pi}{{4}}\right)}}}-{\cos{{\left({x}-\frac{\pi}{{4}}\right)}}}={1}$ Find all solutions of the equation in the interval [0, 2pi).

Is there any personification in "The Tell-Tale Heart"?