BenoguigoliB

2021-08-07

Solving Basic Trigonometric Equations by Factoring Solve the given equation.
$3{\mathrm{sin}}^{2}\theta -7\mathrm{sin}\theta +2=0$

Macsen Nixon

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

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