Trigonometric equation

[eco&math=die]

sin(2x-3)=2/3 find approximate solution(s) for 0 <=x<2pi
I can only get x=1.86 which is 2x-3=0.730 and x=5.01 (2x-3=7.01)
There are still two solutions. I am not sure how to get them. Thanks for helping me

 

[Deleted member 4993]

sin(2x-3)=2/3 find approximate solution(s) for 0 <=x<2pi
I can only get x=1.86 which is 2x-3=0.730 and x=5.01 (2x-3=7.01)
There are still two solutions. I am not sure how to get them. Thanks for helping me

Have you used the fact that:

sin(Θ) = sin(π - Θ)

 

[eco&math=die]

Thanks for the remainder, however, I still get the same results.

 

[BigBeachBanana]

You can solve it in exact form. No need to approximate. Start by taking the arcsin on both sides.

 

[Deleted member 4993]

Thanks for the remainder, however, I still get the same results. View attachment 30599

The second solution should be:

π - 0.73 = 2x - 3

Now solve for 'x' for the second value.

 

[eco&math=die]

The second solution should be:

π - 0.73 = 2x - 3

Now solve for 'x' for the second value.

I get x=2.71, and I know how to find the final x value now, thanks a lot.

 

[blamocur]

sin(2x-3)=2/3 find approximate solution(s) for 0 <=x<2pi
I can only get x=1.86 which is 2x-3=0.730 and x=5.01 (2x-3=7.01)
There are still two solutions. I am not sure how to get them. Thanks for helping me

Why two solutions? When I plot [imath]\sin(2x-3)[/imath] for [imath]0 <= x < 2\pi[/imath] I can see a total of 4 solutions.

 

[BigBeachBanana]

[math]\text{Solution\,1:}\\ 2x-3=\arcsin\left(\frac{2}{3}\right)+2\pi n \Rightarrow x =\frac{1}{2}\left(\arcsin\left(\frac{2}{3}\right)+3\right) +\pi n; n\in [0,1]\\ \text{Solution\,2:}\\ 2x-3=\pi-\arcsin\left(\frac{2}{3}\right) +2\pi n \Rightarrow x= \frac{1}{2}\left(\pi+3-\arcsin\left(\frac{2}{3}\right)\right) + \pi n; n\in [0,1][/math]