Answer:
[tex]\displaystyle \theta=\frac{\pi}{3}+2k\pi,\ \frac{5\pi}{3}+2k\pi,\ \frac{3\pi}{4}+2k\pi,\ \frac{7\pi}{4}+2k\pi[/tex]
Step-by-step explanation:
Trigonometric Equation
A trigonometric equation is any equation that contains one or more trigonometric functions. The solution of a trigonometric equation comes in the form of angles in radians or degrees.
Let's solve the equation
[tex]2cos\theta tan\theta - tan\theta=1-2cos\theta[/tex]
Factoring the left side of the equation by tan[tex]\theta[/tex]
[tex](2cos\theta - 1)tan\theta =1-2cos\theta[/tex]
Rearranging
[tex](2cos\theta - 1)tan\theta +2cos\theta-1=0[/tex]
Factoring again
[tex](2cos\theta - 1)(tan\theta +1)=0[/tex]
We get two separate equations to solve
[tex]\text{[1]\ \ \ }2cos\theta - 1=0[/tex]
[tex]\text{[2]\ \ \ }tan\theta +1=0[/tex]
Solving the first equation
[tex]\displaystyle cos\theta=\frac{1}{2}[/tex]
We get two solutions in the first rotation of [tex]\theta[/tex]
[tex]\displaystyle \theta=\frac{\pi}{3}[/tex]
[tex]\displaystyle \theta=\frac{5\pi}{3}[/tex]
The general solution, being k any integer:
[tex]\displaystyle \theta=\frac{\pi}{3}+2k\pi\\\displaystyle \theta=\frac{5\pi}{3}+2k\pi[/tex]
Solving the second equation
[tex]tan\theta=-1[/tex]
We also get two solutions in the first rotation of [tex]\theta[/tex]
[tex]\displaystyle \theta=\frac{3\pi}{4}[/tex]
[tex]\displaystyle \theta=\frac{7\pi}{4}[/tex]
The general solution, being k any integer:
[tex]\displaystyle \theta=\frac{3\pi}{4}+2k\pi\\\displaystyle \theta=\frac{7\pi}{4}+2k\pi[/tex]
The total solution is
[tex]\displaystyle \theta=\frac{\pi}{3}+2k\pi,\ \frac{5\pi}{3}+2k\pi,\ \frac{3\pi}{4}+2k\pi,\ \frac{7\pi}{4}+2k\pi[/tex]
