SOlve over (2,pie) and find the general solution in radians. 2sin3x = square root of 2
To solve the equation 2sin(3x) = √2 over the interval (0, 2π) and find the general solution in radians, we can follow these steps:
Step 1: Rewrite the equation:
2sin(3x) = √2
Step 2: Divide both sides of the equation by 2:
sin(3x) = √2/2
Step 3: Determine the reference angle:
The reference angle is the angle whose sine value matches √2/2. In this case, the reference angle is π/4 radians or 45 degrees.
Step 4: Finding the solutions:
To find the solutions over the interval (0, 2π), we need to consider the possible values for x that satisfy the equation sin(3x) = √2/2.
The general solution for this equation can be obtained by considering the two half-periods of the sine function.
Half-period 1:
sin(3x) = √2/2
To determine the solutions for this half-period, we solve for x within the range of 0 to π:
3x = π/4
x = π/12
Half-period 2:
sin(3x) = √2/2
To determine the solutions for this half-period, we solve for x within the range of π to 2π:
3x = 3π/4
x = π/4
Thus, the solutions for the equation 2sin(3x) = √2 over the interval (0, 2π) in radians are:
x = π/12
x = π/4
These are the general solutions within the specified interval.