Question

The sum of the three smallest positive values of theta such that 4(cos(theta))(sin(theta)) = 1 is k(pi). Find k.

Answer 1

We are given

$4cos(\theta)sin(\theta)=1$

We can divide both sides by 2

$2cos(\theta)sin(\theta)=\frac{1}{2}$

now, we can use trig identity

$2sin(x)cos(x)=sin(2x)$

so, we can use it

$sin(2\theta)=\frac{1}{2}$

we can take sin^-1 both sides

$2\theta=sin^{-1}(\frac{1}{2} )$

We can take first three positive value

First value is:

$2\theta=\frac{\pi}{2}$

$\theta=\frac{\pi}{4}$

Second value is:

$2\theta=\frac{\pi}{2}+2\pi$

$2\theta=\frac{5\pi}{2}$

$\theta=\frac{5\pi}{4}$

Third value is:

$2\theta=\frac{\pi}{2}+4\pi$

$2\theta=\frac{9\pi}{2}$

$\theta=\frac{9\pi}{4}$

now, we can add them

$\theta=\frac{\pi}{4}+\frac{5\pi}{4}+\frac{9\pi}{4}$

$\theta=\frac{15\pi}{4}$

now, we are given that sum is k(pi)

so, we can set them equal

$k\pi=\frac{15\pi}{4}$

Divide both sides by pi

and we get

$k=\frac{15}{4}$................Answer