Determine solutions for the equation 5sin2x+2sin⁡x=0 in the interval x=[−2π,2π] algebraically. Please round your irrational answers to 3 digits after the decimal point (for example x=2.378 or x=0.001). Please use appropriate notation for your final answer please give me all solutions between [−2π,2π] for this equation.

Question

Determine solutions for the equation 5sin2x+2sin⁡x=0 in the interval x=[−2π,2π] algebraically.   Please round your irrational answers to 3 digits after the decimal point (for example x=2.378 or x=0.001). Please use appropriate notation for your final answer   please give me all solutions between [−2π,2π]  for this equation.

encolatgehu · Accepted Answer

The given trigonometric equation is 5sin2x+2sin⁡x=0 in the interval x∈[−2π,2π]  We have, 5sin2x+2sin⁡x=0  sin⁡x(5sin⁡x+2)=0  sin⁡x=0 or 5sin⁡x+2=0  x=−2π,−π,0,2π or sin⁡x=−25  x=−2π,−π,0,π,2π or x=−0.421,−2.73,3.553,5.872  x=−6.283,−3.142,0,3.142,6.283 or x=−0.421,−2.73,3.553,5.872  ⇒x∈{−6.283,−3.142,−2.73,−0.421,0,3.142,3.553,5.872,6.283}  Hence all solution between [−2π,2π] are x=−6.283,−3.142,−2.730,−0.421,0,3.142,3.553,5.872,6.283.

usumbiix · Accepted Answer

The solution of the equation given by;  5sin2x+2sin⁡x=0  Is the value of the argument of sin function (x) in the question, for which the the equation is satisfied.  Now to solve the above equation, first subtract 2 sin(x) from both side of the equation as follows;  5sin2x+2sin⁡x=0  5sin2x+2sin⁡x−2sin⁡x=0−2sin⁡x  5sin2x=−2sin⁡x (1)  ⇒sin⁡x=0  ∴x=sin−10  =0,±π,±,2π  Now dividing both sides of the equation (1) with 5 sin(x) and solving for x as follows;  5sin2x=−2sin⁡x(1)÷(5sin⁡x)  ⇒5sin2x5sin⁡x=−2sin⁡x5sin⁡x  sin⁡x=−0.4  ∴x=sin−1(−0.4)  =−0.412c  ∵sin⁡(x+2π)=sin⁡x  x=−0.412+2π  =5.871c  ∴x=−0.412c,5.871c  Therefore the solution of the equation in the interval [−2π,2π] is given by;  x=0,±π,±2π,−0.412,5.871 radians

Don Sumner · Accepted Answer

5sin2⁡x+2sin⁡x=0 sin⁡x[5sin⁡x+2]=0 sin⁡x=0 x=−2π,−π,0 π,2π 5sin⁡x+2=0 sin⁡x=−25 x=sin−1⁡−25 x=−sin−1⁡25 x=−2.73,−0412,3.553,5.872 No of solution =9

Determine solutions for the equation 5 \sin^{2} x + 2 \si

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