Notes: `cos(pi/16)cos((3pi)/16) - sin(pi/16)sin((3pi)/16)` Find the exact value of the expression.

Tuesday, 19 April 2016

$\displaystyle{\cos{{\left(\frac{\pi}{{16}}\right)}}}{\cos{{\left(\frac{{{3}\pi}}{{16}}\right)}}}-{\sin{{\left(\frac{\pi}{{16}}\right)}}}{\sin{{\left(\frac{{{3}\pi}}{{16}}\right)}}}$ Find the exact value of the expression.

You need to evaluate the expression using the formula $\displaystyle{\cos{{a}}}\cdot{\cos{{b}}}-{\sin{{a}}}\cdot{\sin{{b}}}={\cos{{\left({a}+{b}\right)}}}$ . You need to put $\displaystyle{a}=\frac{\pi}{{16}}$ and $\displaystyle{b}=\frac{{{3}\pi}}{{16}},$ such that:

$\displaystyle{\cos{{\left(\frac{\pi}{{16}}\right)}}}\cdot{\cos{{\left(\frac{{{3}\pi}}{{16}}\right)}}}-{\sin{{\left(\frac{\pi}{{16}}\right)}}}\cdot{\sin{{\left(\frac{{{3}\pi}}{{16}}\right)}}}={\cos{{\left(\frac{\pi}{{16}}+\frac{{{3}\pi}}{{16}}\right)}}}$

Hence, evaluating the...

Hence, evaluating the given expression yields that it is the cosine of the sum of the angles $\displaystyle{a}=\frac{\pi}{{16}}$ and $\displaystyle{b}=\frac{{{3}\pi}}{{16}}$ , such that $\displaystyle{\cos{{\left(\frac{\pi}{{16}}\right)}}}\cdot{\cos{{\left(\frac{{{3}\pi}}{{16}}\right)}}}-{\sin{{\left(\frac{\pi}{{16}}\right)}}}\cdot{\sin{{\left(\frac{{{3}\pi}}{{16}}\right)}}}={\cos{{\left(\frac{\pi}{{4}}\right)}}}=\frac{\sqrt{{2}}}{{2}}.$