`tan(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.)

Sunday, March 9, 2014

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

Angles u and v are in quadrant 2.

A right triangle can be drawn in quadrant 2. Since $\displaystyle{\sin{{\left({u}\right)}}}=\frac{{5}}{{13}}$ we know that the side opposite angle u is 5 and the hypotenuse of the triangle is 13. Using the pythagorean theorem the third side of the triangle is 12.

A second triangle can be drawn in quadrant 2. Since $\displaystyle{\cos{{\left({v}\right)}}}=-\frac{{3}}{{5}}$ we know that the side adjacent angle v is 3 and the hypotenuse of the triangle is 5. Using the pythagorean theorem the third side of the triangle is 4.

$\displaystyle{\tan{{\left({u}+{v}\right)}}}=\frac{{{\tan{{\left({u}\right)}}}+{\tan{{\left({v}\right)}}}}}{{{1}-{\tan{{\left({u}\right)}}}{\tan{{\left({v}\right)}}}}}$

$\displaystyle{\tan{{\left({u}+{v}\right)}}}=\frac{{{\left(-\frac{{5}}{{12}}\right)}+{\left(-\frac{{4}}{{3}}\right)}}}{{{1}-{\left(-\frac{{5}}{{12}}\right)}{\left(-\frac{{4}}{{3}}\right)}}}=\frac{{\frac{{-{5}+{\left(-{16}\right)}}}{{12}}}}{{{1}-{\left(\frac{{20}}{{36}}\right)}}}=\frac{{-\frac{{21}}{{12}}}}{{\frac{{16}}{{36}}}}=-\frac{{63}}{{16}}$

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Sunday, March 9, 2014

$\displaystyle{\tan{{\left({u}+{v}\right)}}}$ 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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