Notes: `sin(pi/12)cos(pi/4) + cos(pi/12)sin(pi/4)` Find the exact value of the expression.

Monday, 4 January 2016

$\displaystyle{\sin{{\left(\frac{\pi}{{12}}\right)}}}{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\cos{{\left(\frac{\pi}{{12}}\right)}}}{\sin{{\left(\frac{\pi}{{4}}\right)}}}$ Find the exact value of the expression.

You need to recognize the formula sin(a+b) = sin a*cos b + sin b*cos a. You need to put $\displaystyle{a}=\frac{\pi}{{12}}$ and $\displaystyle{b}=\frac{\pi}{{4}}$ , such that:

$\displaystyle{\sin{{\left(\frac{\pi}{{12}}+\frac{\pi}{{4}}\right)}}}={\sin{{\left(\frac{\pi}{{12}}\right)}}}\cdot{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left(\frac{\pi}{{12}}\right)}}}$

$\displaystyle{\sin{{\left({4}\frac{\pi}{{12}}\right)}}}={\sin{{\left(\frac{\pi}{{12}}\right)}}}\cdot{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left(\frac{\pi}{{12}}\right)}}}$

$\displaystyle{\sin{{\left(\frac{\pi}{{3}}\right)}}}={\sin{{\left(\frac{\pi}{{12}}\right)}}}\cdot{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left(\frac{\pi}{{12}}\right)}}}$

$\displaystyle{\sin{{\left(\frac{\pi}{{12}}\right)}}}\cdot{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left(\frac{\pi}{{12}}\right)}}}=\frac{\sqrt{{3}}}{{2}}$

Hence,...

You need to recognize the formula sin(a+b) = sin a*cos b + sin b*cos a. You need to put $\displaystyle{a}=\frac{\pi}{{12}}$ and $\displaystyle{b}=\frac{\pi}{{4}}$ , such that:

$\displaystyle{\sin{{\left(\frac{\pi}{{12}}\right)}}}\cdot{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left(\frac{\pi}{{12}}\right)}}}=\frac{\sqrt{{3}}}{{2}}$

Hence, the given expression could be evaluated as the sine of the sum of angles $\displaystyle\frac{\pi}{{12}}$ and $\displaystyle\frac{\pi}{{4}},$ such that $\displaystyle{\sin{{\left(\frac{\pi}{{3}}\right)}}}={\sin{{\left(\frac{\pi}{{12}}\right)}}}\cdot{\cos{{\left(\frac{\pi}{{4}}\right)}}}+{\sin{{\left(\frac{\pi}{{4}}\right)}}}{\cos{{\left(\frac{\pi}{{12}}\right)}}}=\frac{\sqrt{{3}}}{{2}}$