`sin(u) = -3/5, (3pi)/2

Thursday, August 11, 2016

$\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{3}}{{5}},\frac{{{3}\pi}}{{2}}$

Given $\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{3}}{{5}},\frac{{{3}\pi}}{{2}}<{u}<{2}\pi$

Angle u is in quadrant 4. A right triangle is drawn in quadrant 4. Since $\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{3}}{{5}}$

the side opposite from angle u is 3 and the hypotenuse is 5. Using the pythagorean theorem, the side adjacent to angle u is 4.

$\displaystyle{\sin{{\left({2}{u}\right)}}}={2}{\sin{{\left({u}\right)}}}{\cos{{\left({u}\right)}}}={2}{\left(-\frac{{3}}{{5}}\right)}{\left(\frac{{4}}{{5}}\right)}=-\frac{{24}}{{25}}$

$\displaystyle{\cos{{\left({2}{u}\right)}}}={1}-{2}{{\sin}}^{{2}}{\left({u}\right)}={1}-{2}{{\left(-\frac{{3}}{{5}}\right)}}^{{2}}={1}-{2}{\left(\frac{{9}}{{25}}\right)}=\frac{{25}}{{25}}-\frac{{18}}{{25}}=\frac{{7}}{{25}}$

$\displaystyle{\tan{{\left({2}{u}\right)}}}={2}\frac{{\tan{{\left({u}\right)}}}}{{{1}-{{\tan}}^{{2}}{\left({u}\right)}}}={2}\frac{{-\frac{{3}}{{4}}}}{{{1}-{{\left(-\frac{{3}}{{4}}\right)}}^{{2}}}}=\frac{{-\frac{{3}}{{2}}}}{{{1}-\frac{{9}}{{16}}}}=\frac{{-\frac{{3}}{{2}}}}{{\frac{{16}}{{16}}-\frac{{9}}{{16}}}}=\frac{{-\frac{{3}}{{2}}}}{{\frac{{7}}{{16}}}}=-\frac{{24}}{{7}}$

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Thursday, August 11, 2016

$\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{3}}{{5}},\frac{{{3}\pi}}{{2}}$

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What was the device called which Faber had given Montag in order to communicate with him?