Question

Consider the equation 1 - 3 tan^2x = 0. Find all the values of x in the interval (0,2pi) that satisfy the equation.

A) x = π/4
B) x = 3π/4
C) x = 5π/4
D) x = 7π/4
E) x = π/2, 3π/2

Answer 1

Solving 1 - 3 tan^2x = 0 gives tanx = ±√(1/3), and since this value does not correspond to the provided options, none of the provided answer choices are correct.

Step-by-step explanation:

First, solve the trigonometric equation 1 - 3 tan^2x = 0. Setting 3 tan^2x equal to 1 yields tan^2x = 1/3. Taking the square root of both sides (and remembering the ± for the square root of a term), we find that tanx = ±√(1/3). To determine all values of x in the interval (0,2π) where tangent has the values ±√(1/3), we observe that tangent is positive in the first and third quadrants and negative in the second and fourth quadrants.

Since the square root of 1/3 does not correspond to any of the well-known angle values for tangent, we can conclude that none of the provided multiple-choice answers A) π/4, B) 3π/4, C) 5π/4, D) 7π/4, or E) π/2, 3π/2 are correct because these angles correspond to tangent values of ±1 or undefined, not ±√(1/3).