How to determine the solutions?

Question

Determine the solution(s) to the equation root3tantheta + 3 = 1 where 0</=theta</=2pi.

Answer 1

√3tanθ+3=1

√3tanθ=-2               (add 3 both sides)

tanθ=-2/√3              (divide √3 both sides)

tanθ=-2√3/3            (rationalize by multiplying √3/√3)

tan^-1(-2√3/3)=θ      (convert to inverse tangent function)

 

Since this is ratio is not among the standard family, we will have to enter tan-1(-2√3/3) into the calculator. 

We get -.857, or -.273π.

For tangent we must use the rule that tan(a)=tan(a+pi) to find all possible solutions.

-.273π+π = .727π

1.27π+π = 1.727π

These are the two answers that fit within the range of 0<θ<2π.

 

Thus our solution set is θ=.727π, 1.727π in terms of pi

or θ=2.284, 5.426 in decimal form.