`csc(u - v)` Find the exact value of the trigonometric expression given that sin(u) = 5/13 and cos(v) = -3/5 (both u and v are in quadrant II.)

Thursday, September 2, 2010

$\displaystyle{\csc{{\left({u}-{v}\right)}}}$ Find the exact value of the trigonometric expression given that sin(u) = 5/13 and cos(v) = -3/5 (both u and v are in quadrant II.)

Given $\displaystyle{\sin{{\left({u}\right)}}}=\frac{{5}}{{13}},{\cos{{\left({v}\right)}}}=-\frac{{3}}{{5}}$

using pythagorean identity,

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

$\displaystyle{{\left(\frac{{5}}{{13}}\right)}}^{{2}}+{{\cos}}^{{2}}{\left({u}\right)}={1}$

$\displaystyle{{\cos}}^{{2}}{\left({u}\right)}={1}-\frac{{25}}{{169}}=\frac{{{169}-{25}}}{{169}}=\frac{{144}}{{169}}$

$\displaystyle{\cos{{\left({u}\right)}}}=\sqrt{{\frac{{144}}{{169}}}}$

$\displaystyle{\cos{{\left({u}\right)}}}=\pm\frac{{12}}{{13}}$

since u is in quadrant II ,

$\displaystyle\therefore{\cos{{\left({u}\right)}}}=-\frac{{12}}{{13}}$

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

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+{{\left(-\frac{{3}}{{5}}\right)}}^{{2}}={1}$

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

$\displaystyle{\sin{{\left({v}\right)}}}=\sqrt{{\frac{{16}}{{25}}}}$

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

since v is in quadrant II,

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

Now let's evaluate csc(u-v),

$\displaystyle{\csc{{\left({u}-{v}\right)}}}=\frac{{1}}{{\sin{{\left({u}-{v}\right)}}}}$

$\displaystyle=\frac{{1}}{{{\sin{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}-{\cos{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}}}$

plug in the values of sin(u),cos(v),cos(u) and sin(v),

$\displaystyle=\frac{{1}}{{{\left(\frac{{5}}{{13}}\right)}{\left(-\frac{{3}}{{5}}\right)}-{\left(-\frac{{12}}{{13}}\right)}{\left(\frac{{4}}{{5}}\right)}}}$

$\displaystyle=\frac{{1}}{{-\frac{{3}}{{13}}+\frac{{48}}{{65}}}}$

$\displaystyle=\frac{{65}}{{33}}$

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Thursday, September 2, 2010

$\displaystyle{\csc{{\left({u}-{v}\right)}}}$ Find the exact value of the trigonometric expression given that sin(u) = 5/13 and cos(v) = -3/5 (both u and v are in quadrant II.)

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