Question

Solve this equation in the

given domain: 0 *give answer in radians*
3 sin (2x-4)=2
There should be 4 answers.

look at attached photo

Answer 1

`3\sin(2x-4)=2\implies \sin(2x-4)=\cfrac{2}{3}\implies 2x-4=\sin^(-1)\left( \cfrac{2}{3} \right) \\\\\\ 2x=\sin^(-1)\left( \cfrac{2}{3} \right)+4\implies x=\cfrac{\sin^(-1)\left( (2)/(3) \right)+4}{2} \\\\\\ x=\cfrac{\sin^(-1)\left( (2)/(3) \right)}{2}+2\implies x\approx\cfrac{0.73}{2}+2\implies \stackrel{II~Quadrant}{x\approx 2.365}\hspace{5em}\stackrel{III~Quadrant}{\stackrel{(\pi -0.73)/(2)+2}{x\approx 3.206}}`

Answer 2

Answer: 2.294524311274043, 4.294524311274043, 6.294524311274043 and 8.294524311274043 radians.

Step-by-step explanation: We can start by isolating the sin function on one side of the equation:

sin (2x-4) = 2/3

Next, we can use the inverse sine function (sin^-1) to find the value of 2x-4:

2x-4 = sin^-1(2/3)

We know that the domain of the inverse sine function is [-1,1], so the value of 2/3 falls in that range. To find the value of x, we can add 4 to both sides and divide both sides by 2:

x = (sin^-1(2/3) + 4) / 2

The value of sin^-1(2/3) is approximately 0.5890486225480862 radians and that's the first solution in the range [0, pi], and we can add multiples of pi to get the other solutions.

x = (0.5890486225480862 + 4) / 2 = 2.294524311274043 radians (in the range [0, pi])

x = (0.5890486225480862 + 4 + pi) / 2 = 4.294524311274043 radians (in the range [pi, 2pi])

x = (0.5890486225480862 + 4 + 2pi) / 2 = 6.294524311274043 radians (in the range [2pi, 3pi])

x = (0.5890486225480862 + 4 + 3pi) / 2 = 8.294524311274043 radians (in the range [3pi, 4*pi])

So there are four solutions to this equation: 2.294524311274043, 4.294524311274043, 6.294524311274043 and 8.294524311274043 radians.