`cos(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, January 22, 2015

$\displaystyle{\cos{{\left({u}-{v}\right)}}}$ 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 $\displaystyle{\sin{{\left({u}\right)}}}=\frac{{5}}{{13}}$ you know that the side opposite of angle u is 5 and the hypotenuse is 13. Using the pythagorean theorem the third side of the triangle is 12.

A right triangle can be drawn in quadrant 2 since $\displaystyle{\cos{{\left({v}\right)}}}=-\frac{{3}}{{5}}$ you know

that the side adjacent to angle v is 3 and the hypotenuse is 5. Using the pythagorean theorem the third side of the triangle is 4.

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

$\displaystyle{\cos{{\left({u}-{v}\right)}}}={\left(-\frac{{12}}{{13}}\right)}{\left(-\frac{{3}}{{5}}\right)}+{\left(\frac{{5}}{{13}}\right)}{\left(\frac{{4}}{{5}}\right)}=\frac{{36}}{{65}}+\frac{{20}}{{65}}=\frac{{56}}{{65}}$

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Thursday, January 22, 2015

$\displaystyle{\cos{{\left({u}-{v}\right)}}}$ 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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