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

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

Macsen Nixon · Accepted Answer

Approach:  The domain of the trigonometry function of sin⁡θ is lies between [-1,1]. No solution exists beyond this domain. Sine has period 2π, we find solution in any interval of length 2π. SIne function is positive in first and second quadrant.  Calculation:  Consider the trigonometry equation.  3sin2θ−7sin⁡θ+2=0  The factor of the given equation is obtained by,  3sin2θ−7sin⁡θ+2=0  (sin⁡θ−2)(3sin⁡θ−1)=0  The factors of above equation are,  sin⁡θ−2=0……(1)  3sin⁡θ−1=0……(2)  The solution obtained for the factor in which sine function involved so we will get the solution in the interval of [0,2π].  Consider equation (1).  sin⁡θ−2=0  The above equation has no solution because value lies outside the domain.  Add 1 both sides in equation (2).  3sin⁡θ−1=0  θ=sin1(13)  θ=0.34,2.80  Here, the angles are in radian.  The sine has period, 2π So we get all solutions of the equation by adding integer multiples of 2π to these solutions: S θ=0.34+2kπ  θ=2.80+2kπ  Therefore, the solutions of the trigonometry equation 3sin2θ−7sin⁡θ+2=0 are θ=0.34+2kπ and θ=2.80+2kπ.  Concusion:  Thus, the solutions of the trigonometry equation 3sin2θ−7sin⁡θ+2=0 are θ=0.34+2kπ and θ=2.80+2kπ.

Solving Basic Trigonometric Equations by Factoring Solve the given equation. 3

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