How to find all solutions for 3tan^22x-1=0?
To find all the solutions for the equation 3tan^2(2x) - 1 = 0, you will need to follow these steps:
Step 1: Rearrange the equation
Start by adding 1 to both sides of the equation to remove the -1 term:
3tan^2(2x) = 1
Step 2: Divide both sides by 3
Divide both sides of the equation by 3:
tan^2(2x) = 1/3
Step 3: Take the square root of both sides
Take the square root of both sides of the equation:
tan(2x) = ±√(1/3)
Step 4: Find the tangent's angle
Use the inverse tangent (arctan) function to find the angles that have a tangent equal to √(1/3) or -√(1/3):
2x = arctan(√(1/3)) + nπ or 2x = arctan(-√(1/3)) + nπ
where n is an integer representing additional solutions due to periodicity.
Step 5: Solve for x
Divide both sides of the equation by 2 to solve for x:
x = (arctan(√(1/3)) + nπ)/2 or x = (arctan(-√(1/3)) + nπ)/2
These are the solutions for the equation 3tan^2(2x) - 1 = 0, where n is an integer. Be cautious of any restrictions or specific ranges stated in the original problem statement, as it might affect the possible values of x.