Product of Sine and Cosine

Geometry Level 2

$\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{\pi}{8}\right) = \frac{a\sqrt{b}}{c}$ , where $a, b$ and $c$ are integers such that $a$ and $c$ are coprime, and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$ ?

Details and assumptions

$a, b$ and $c$ are allowed to be 1. In particular, if you think the value is $1 = \frac {1 \sqrt{1} } {1}$ , then $a+b+c=1+1+1=3$ .

The answer is 7.

1 solution

Arron Kau Staff
May 13, 2014

Using the double angle formula $\sin(2x) = 2\sin(x)\cos(x)$ we have:

$\sin\left(\frac{\pi}{8}\right)\cos\left(\frac{\pi}{8}\right) = \frac{\sin\left(\frac{\pi}{4}\right)}{2} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}$ .

Hence $a + b + c= 1 +2 + 4= 7$ .

Note: Students were not expected to evaluate $\sin\left(\frac{\pi}{8}\right)$ or $\cos\left(\frac{\pi}{8}\right)$ .

Product of Sine and Cosine

The answer is 7.

1 solution

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