How do I solve, in radians and degrees, 2-3sin(3(x-pi))=6?
To solve the equation 2 - 3sin(3(x - π)) = 6, you can follow these steps:
Step 1: Simplify the equation.
First, start by subtracting 2 from both sides of the equation:
2 - 3sin(3(x - π)) - 2 = 6 - 2
-3sin(3(x - π)) = 4
Step 2: Isolate the trigonometric function.
Next, divide both sides of the equation by -3:
-3sin(3(x - π)) / (-3) = 4 / (-3)
sin(3(x - π)) = -4/3
Step 3: Solve for the argument.
Since we are looking for both radians and degrees of the solution, we need to solve for the argument of the sine function: 3(x - π).
sin(3(x - π)) = -4/3
Step 4: Find the reference angle.
To find the reference angle, take the inverse sine of the absolute value of -4/3:
Reference angle = arcsin(abs(-4/3))
Step 5: Solve for x in radians.
To find the values of x in radians, we can set up two equations since sin has a period of 2π:
3(x - π) = arcsin(4/3) + 2πn
3(x - π) = π - arcsin(4/3) + 2πm
(where n and m are integers)
Now solve for x:
x = (arcsin(4/3) + 2πn)/3 + π
x = (π - arcsin(4/3) + 2πm)/3 + π
These equations will give you all possible solutions in radians.
Step 6: Solve for x in degrees.
To find the values of x in degrees, convert the radians obtained in Step 5 to degrees by multiplying by 180/π:
x_degrees = x * (180/π)
Evaluate the equations obtained in Step 5 in degrees to find all possible solutions.
Note: Keep in mind that trigonometric equations can have an infinite number of solutions due to their periodic nature, so it's important to include the correct notations for the values of n and m, which represent integers.