We are given
[tex]4cos(\theta)sin(\theta)=1[/tex]
We can divide both sides by 2
[tex]2cos(\theta)sin(\theta)=\frac{1}{2}[/tex]
now, we can use trig identity
[tex]2sin(x)cos(x)=sin(2x)[/tex]
so, we can use it
[tex]sin(2\theta)=\frac{1}{2}[/tex]
we can take sin^-1 both sides
[tex]2\theta=sin^{-1}(\frac{1}{2} )[/tex]
We can take first three positive value
First value is:
[tex]2\theta=\frac{\pi}{2}[/tex]
[tex]\theta=\frac{\pi}{4}[/tex]
Second value is:
[tex]2\theta=\frac{\pi}{2}+2\pi[/tex]
[tex]2\theta=\frac{5\pi}{2}[/tex]
[tex]\theta=\frac{5\pi}{4}[/tex]
Third value is:
[tex]2\theta=\frac{\pi}{2}+4\pi[/tex]
[tex]2\theta=\frac{9\pi}{2}[/tex]
[tex]\theta=\frac{9\pi}{4}[/tex]
now, we can add them
[tex]\theta=\frac{\pi}{4}+\frac{5\pi}{4}+\frac{9\pi}{4}[/tex]
[tex]\theta=\frac{15\pi}{4}[/tex]
now, we are given that sum is k(pi)
so, we can set them equal
[tex]k\pi=\frac{15\pi}{4}[/tex]
Divide both sides by pi
and we get
[tex]k=\frac{15}{4}[/tex]................Answer
