Solutions of trigonometric equations

[lionel messi.]

1. The problem statement:
QUESTION:the general solution of equation (√3 - 1)sin∅+(√3 + 1)c0s∅=2 is
1) ∅=2nπ+- (π/4)+(π/12)
2) ∅= nπ + (-1)^π (π/4)+(π/12)
3) ∅=2(n+1)π (π/4)+(π/12)

The Attempt at a Solution

ACTUALLY I DONT NEED THE ATTEMPT, i know the answer is option 1 , i don't know why the last 2 options can be the solutions of the question as well.my teacher told me that to use 2nπ with sin and cos functions and nπ with tan functions.can someone please describe in detail about the periods used in solutions of trig.equations.THANKS

 

[LCKurtz]

If you have a solution $\theta_0$ to an equation like $\sin\theta = c$, then since the sine function has period $2\pi$, for any integer multiple like $2n\pi$ you will have $\sin(\theta_0+2n\pi)=c$ also. That's why you add the $2n\pi$ to the $\theta_0$ to get all the solutions. The same idea for tangent or cotangent applies, except since they have periods of $\pi$, you add multiples $n\pi$ to your solution.