`cos(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.)

Sunday, March 8, 2009

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

Angles u and v are in quadrant 3.

A right triangle can be drawn in quadrant 3. Since $\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{7}}{{25}}$ we know that the side opposite angle u is 7 and the hypotenuse is 25. Using the pythagorean theorem the third side of the triangle is 24.

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

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

$\displaystyle{\cos{{\left({u}+{v}\right)}}}={\left(-\frac{{24}}{{25}}\right)}{\left(-\frac{{4}}{{5}}\right)}-{\left(-\frac{{7}}{{25}}\right)}{\left(-\frac{{3}}{{5}}\right)}={\left(\frac{{96}}{{125}}\right)}-{\left(\frac{{21}}{{25}}\right)}=\frac{{75}}{{125}}=\frac{{3}}{{5}}$

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Sunday, March 8, 2009

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

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