Solve the equation.
(tan^2)3x=3
To solve the equation (tan^2)3x = 3, we can follow these steps:
Step 1: Rewrite the equation
(tan^2)3x = 3
Step 2: Take the square root of both sides
√[(tan^2)3x] = √3
Step 3: Simplify the square root
|tan(3x)| = √3
Step 4: Determine the possible values for tan(3x)
tan(3x) could be either positive or negative square root of 3.
Step 5: Solve for x
To find the values of x that satisfy the equation, we need to solve for 3x.
If tan(3x) = √3:
By taking the inverse tangent (tan^-1) of both sides, we get:
3x = tan^-1(√3)
x = (1/3) * tan^-1(√3)
If tan(3x) = -√3:
Similarly, by taking the inverse tangent (tan^-1) of both sides, we get:
3x = tan^-1(-√3)
x = (1/3) * tan^-1(-√3)
So, the solution for the equation (tan^2)3x = 3 is:
x = (1/3) * tan^-1(√3), (1/3) * tan^-1(-√3)
Tan^2(3x) = 3
Apply square root on both sides of the equation
tan 3x = + sqrt(3) or -sqrt(3)= tan(π /3) or - tan(π/3)
Apply inverse tangent to both sides of equation
3x = π/3 + kπ or -π/3 + kπ where k is integer
Divide both sides by 3
x = π/9 + kπ/3 or -π/9 + kπ/3 which implies
x = ±π/9 + kπ/3