`sin(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...

Tuesday, September 16, 2008

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

We know that $\displaystyle{\sin{{\left({u}+{v}\right)}}}={\sin{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}+{\cos{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}.$

$\displaystyle{\sin{{\left({u}\right)}}}$ and $\displaystyle{\cos{{\left({v}\right)}}}$ are given, let's find $\displaystyle{\cos{{\left({u}\right)}}}$ and $\displaystyle{\sin{{\left({v}\right)}}}.$

$\displaystyle{\cos{{\left({u}\right)}}}=\pm\sqrt{{{1}-{{\sin}}^{{2}}{\left({u}\right)}}}=\pm\sqrt{{{1}-\frac{{25}}{{169}}}}=\pm\sqrt{{\frac{{144}}{{169}}}}=\pm\frac{{12}}{{13}}.$

Select "-" because $\displaystyle{u}$ is in quadrant II.

And $\displaystyle{\sin{{\left({v}\right)}}}=\pm\sqrt{{{1}-{{\cos}}^{{2}}{\left({v}\right)}}}=\pm\sqrt{{{1}-\frac{{9}}{{25}}}}=\pm\frac{{4}}{{5}}.$

Select "+" because $\displaystyle{v}$ is in quadrant II.

Finally $\displaystyle{\sin{{\left({u}+{v}\right)}}}={\left(\frac{{5}}{{13}}\right)}\cdot{\left(-\frac{{3}}{{5}}\right)}+{\left(-\frac{{12}}{{13}}\right)}\cdot{\left(\frac{{4}}{{5}}\right)}=-\frac{{1}}{{65}}\cdot{\left({15}+{48}\right)}=-\frac{{63}}{{65}}.$

Blog

Tuesday, September 16, 2008

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

No comments:

Post a Comment

What was the device called which Faber had given Montag in order to communicate with him?

Tuesday, September 16, 2008

Find the exact value of the trigonometric expression given that and (both and are in quadrant...

No comments:

Post a Comment

What was the device called which Faber had given Montag in order to communicate with him?

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