Question

Super urgent! Solve the equation for all real values of x. 3tan^2 = sqrt3 tanx

Answer 1

f(x) = tan2(x) + (√3 - 1)[tan(x)] - √3 = 0

tan2(x) + √3[tan(x)] - tan(x) - √3 = 0

Factor into

[-1 + tan(x)]*[√3 + tan(x)] = 0

which means

[-1 + tan(x)] = 0 and/or [√3 + tan(x)] = 0

Then

tan(x) = 1

tan-1(1) = pi/4 radians

For the other equation

[√3 + tan(x)] = 0

tan(x) = -√3

tan-1(-√3) = -pi/3

so that

x = pi/4 or -pi/3 in the interval [0, 2pi]