Solving Basic Trigonometric Equations by Factoring Solve the given equation. 3sin2θ−7sin⁡θ+2=0

Approach:
The domain of the trigonometry function of ${\displaystyle \sin \theta }$ is lies between [-1,1]. No solution exists beyond this domain. Sine has period ${\displaystyle 2\pi }$, we find solution in any interval of length ${\displaystyle 2\pi }$. SIne function is positive in first and second quadrant.
Calculation:
Consider the trigonometry equation.
${\displaystyle 3{\sin }^2\theta -7\sin \theta +2=0}$
The factor of the given equation is obtained by,
${\displaystyle 3{\sin }^2\theta -7\sin \theta +2=0}$
${\displaystyle (\sin \theta -2)(3\sin \theta -1)=0}$
The factors of above equation are,
${\displaystyle \sin \theta -2=0\dots \dots \left(1\right)}$
${\displaystyle 3\sin \theta -1=0\dots \dots \left(2\right)}$
The solution obtained for the factor in which sine function involved so we will get the solution in the interval of ${\displaystyle [0,2\pi ]}$.
Consider equation (1).
${\displaystyle \sin \theta -2=0}$
The above equation has no solution because value lies outside the domain.
Add 1 both sides in equation (2).
${\displaystyle 3\sin \theta -1=0}$
${\displaystyle \theta ={\sin }^1\left(\frac{1}{3}\right)}$
${\displaystyle \theta =0.34,2.80}$
Here, the angles are in radian.
The sine has period, ${\displaystyle 2\pi }$ So we get all solutions of the equation by adding integer multiples of ${\displaystyle 2\pi }$ to these solutions:
S ${\displaystyle \theta =0.34+2k\pi }$
${\displaystyle \theta =2.80+2k\pi }$
Therefore, the solutions of the trigonometry equation ${\displaystyle 3{\sin }^2\theta -7\sin \theta +2=0}$ are ${\displaystyle \theta =0.34+2k\pi }$ and ${\displaystyle \theta =2.80+2k\pi }$.
Concusion:
Thus, the solutions of the trigonometry equation ${\displaystyle 3{\sin }^2\theta -7\sin \theta +2=0}$ are ${\displaystyle \theta =0.34+2k\pi }$ and ${\displaystyle \theta =2.80+2k\pi }$.