Answer:
The number of possible solution of the trignometric equation are:
12
Step-by-step explanation:
We have to find the solution of the trignometric equation which is given as:
[tex]\cos(6x)=\dfrac{1}{2}[/tex]
Now, the solution of the trignometric equation is the possible value of x such that the equation holds true.
Now we know that:
[tex]\cos(\dfrac{\pi}{3})=\dfrac{1}{2}[/tex]
Also,
[tex]\cos(2\pi-\dfrac{\pi}{3})=\dfrac{1}{2}\\\\i.e.\\\\\cos(\dfrac{5\pi}{3})=\dfrac{1}{3}[/tex]
We are given that:
0<x<2π.
this means that:
0<6x<12π.
Now, we have the solution as:
[tex]6x=\dfrac{\pi}{3}\\\\\\i.e\\\\x=\dfrac{\pi}{18}[/tex]
and:
[tex]6x=\dfrac{5\pi}{3}\\\\x=\dfrac{5\pi}{18}[/tex]
[tex]6x=2\pi+\dfrac{\pi}{3}=\dfrac{7\pi}{3}\\\\6x=4\pi-\dfrac{\pi}{3}=\dfrac{11\pi}{3}\\\\6x=4\pi+\dfrac{\pi}{3}=\dfrac{13\pi}{3}\\\\6x=6\pi-\dfrac{\pi}{3}=\dfrac{17\pi}{3}\\\\6x=6\pi+\dfrac{\pi}{3}=\dfrac{19\pi}{3}\\\\6x=8\pi-\dfrac{\pi}{3}=\dfrac{23\pi}{3}\\\\6x=8\pi+\dfrac{\pi}{3}=\dfrac{25\pi}{3}\\\\6x=10\pi-\dfrac{\pi}{3}=\dfrac{29\pi}{3}\\\\6x=10\pi+\dfrac{\pi}{3}=\dfrac{31\pi}{3}\\\\6x=12\pi-\dfrac{\pi}{3}=\dfrac{35\pi}{3}[/tex]
Hence, the number of possible solution of the trignometric equation are:
12
