`sin(u + v)` Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

Monday, September 28, 2015

$\displaystyle{\sin{{\left({u}+{v}\right)}}}$ Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

Given $\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{7}}{{25}},{\cos{{\left({v}\right)}}}=-\frac{{4}}{{5}}$

using pythagorean identity,

$\displaystyle{{\sin}}^{{2}}{\left({u}\right)}+{{\cos}}^{{2}}{\left({u}\right)}={1}$

plug in the value of sin(u),

$\displaystyle{{\left(-\frac{{7}}{{25}}\right)}}^{{2}}+{{\cos}}^{{2}}{\left({u}\right)}={1}$

$\displaystyle\frac{{49}}{{625}}+{{\cos}}^{{2}}{\left({u}\right)}={1}$

$\displaystyle{{\cos}}^{{2}}{\left({u}\right)}={1}-\frac{{49}}{{625}}$

$\displaystyle{{\cos}}^{{2}}{\left({u}\right)}=\frac{{{625}-{49}}}{{625}}$

$\displaystyle{{\cos}}^{{2}}{\left({u}\right)}=\frac{{576}}{{625}}$

$\displaystyle{\cos{{\left({u}\right)}}}=\sqrt{{\frac{{576}}{{625}}}}$

$\displaystyle{\cos{{\left({u}\right)}}}=\pm\frac{{24}}{{25}}$

Since u is in Quadrant III ,

$\displaystyle\therefore{\cos{{\left({u}\right)}}}=-\frac{{24}}{{25}}$

Now $\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+{{\cos}}^{{2}}{\left({v}\right)}={1}$

plug in the value of cos(v)=-4/5,

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+{{\left(-\frac{{4}}{{5}}\right)}}^{{2}}={1}$

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+\frac{{16}}{{25}}={1}$

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}={1}-\frac{{16}}{{25}}=\frac{{9}}{{25}}$

$\displaystyle{\sin{{\left({v}\right)}}}=\sqrt{{\frac{{9}}{{25}}}}$

$\displaystyle{\sin{{\left({v}\right)}}}=\pm\frac{{3}}{{5}}$

since v is in Quadrant III ,

$\displaystyle\therefore{\sin{{\left({v}\right)}}}=-\frac{{3}}{{5}}$

$\displaystyle{\sin{{\left({u}+{v}\right)}}}={\sin{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}+{\cos{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

plug in the values ,

$\displaystyle{\sin{{\left({u}+{v}\right)}}}={\left(-\frac{{7}}{{25}}\right)}{\left(-\frac{{4}}{{5}}\right)}+{\left(-\frac{{24}}{{25}}\right)}{\left(-\frac{{3}}{{5}}\right)}$

$\displaystyle{\sin{{\left({u}+{v}\right)}}}=\frac{{28}}{{125}}+\frac{{72}}{{125}}=\frac{{100}}{{125}}$

$\displaystyle{\sin{{\left({u}+{v}\right)}}}=\frac{{4}}{{5}}$

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