# How do you express sin(pi/ 8 ) * cos(( ( 5 pi) / 3 )  without using products of trigonometric functions?

Dec 9, 2017

The answer is $= \frac{1}{2} \left(\sin \left(\frac{1}{3} \pi\right) - \sin \left(\frac{5}{24} \pi\right)\right)$

#### Explanation:

Apply the formula

$\sin a \cos b = \frac{1}{2} \left(\sin \left(a + b\right) + \sin \left(a - b\right)\right)$

You can easily prove this formula by using $\sin \left(a + b\right)$ and $\sin \left(a - b\right)$

Here,

$a = \frac{1}{8} \pi$ and $b = \frac{5}{3} \pi$

Therefore,

$\sin \left(\frac{1}{8} \pi\right) \cos \left(\frac{5}{3} \pi\right) = \frac{1}{2} \left(\sin \left(\frac{1}{8} \pi + \frac{5}{3} \pi\right) + \sin \left(\frac{1}{8} \pi - \frac{5}{3} \pi\right)\right)$

$= \frac{1}{2} \left(\sin \left(\frac{43}{24} \pi\right) + \sin \left(- \frac{37}{24} \pi\right)\right)$

$= \frac{1}{2} \left(\sin \left(- \frac{5}{24} \pi\right) + \sin \left(\frac{8}{24} \pi\right)\right)$

$= \frac{1}{2} \left(\sin \left(\frac{1}{3} \pi\right) - \sin \left(\frac{5}{24} \pi\right)\right)$

Dec 9, 2017

Simply calculate the given terms.
$= 0.00682$

#### Explanation:

You "using products of trig functions" statement is not clear. I took it to mean revising the expression to form a single trigonometric identity.

Thus, to avoid that, I simply evaluated the expression by each term.
sin(π/8) xx cos(5π/3) = 0.00685 xx 0.9958 = 0.00682