Notes: `sin(60) cos(15) + cos(60)sin(15)` Write the expression as a sine, cosine, or tangent of an angle.

Saturday, 17 December 2016

$\displaystyle{\sin{{\left({60}\right)}}}{\cos{{\left({15}\right)}}}+{\cos{{\left({60}\right)}}}{\sin{{\left({15}\right)}}}$ Write the expression as a sine, cosine, or tangent of an angle.

You need to recognize the formula $\displaystyle{\sin{{\left({a}+{b}\right)}}}={\sin{{a}}}\cdot{\cos{{b}}}+{\sin{{b}}}\cdot{\cos{{a}}}$ . You need to put $\displaystyle{a}={{60}}^{{o}}$ and $\displaystyle{b}={{15}}^{{o}}$ , such that:

$\displaystyle{\sin{{\left({{60}}^{{o}}+{{15}}^{{o}}\right)}}}={{\sin{{60}}}}^{{o}}\cdot{{\cos{{15}}}}^{{o}}+{{\sin{{15}}}}^{{o}}\cdot{{\cos{{60}}}}^{{o}}$

$\displaystyle{{\sin{{75}}}}^{{o}}={{\sin{{60}}}}^{{o}}\cdot{{\cos{{15}}}}^{{o}}+{{\sin{{15}}}}^{{o}}\cdot{{\cos{{60}}}}^{{o}}$

Hence, the given expression could be evaluated as the sine of the sum of angles $\displaystyle{{60}}^{{o}}$ and $\displaystyle{{15}}^{{o}}$ ,...

$\displaystyle{\sin{{\left({{60}}^{{o}}+{{15}}^{{o}}\right)}}}={{\sin{{60}}}}^{{o}}\cdot{{\cos{{15}}}}^{{o}}+{{\sin{{15}}}}^{{o}}\cdot{{\cos{{60}}}}^{{o}}$

$\displaystyle{{\sin{{75}}}}^{{o}}={{\sin{{60}}}}^{{o}}\cdot{{\cos{{15}}}}^{{o}}+{{\sin{{15}}}}^{{o}}\cdot{{\cos{{60}}}}^{{o}}$

Hence, the given expression could be evaluated as the sine of the sum of angles $\displaystyle{{60}}^{{o}}$ and $\displaystyle{{15}}^{{o}}$ , such that $\displaystyle{{\sin{{60}}}}^{{o}}\cdot{{\cos{{15}}}}^{{o}}+{{\sin{{15}}}}^{{o}}\cdot{{\cos{{60}}}}^{{o}}={{\sin{{75}}}}^{{o}}.$