Find x where 0 ≤ x ≤ 2pi

Question

4sqrt(2) tan x - sqrt(2)= 3sqrt(2) tan x

Enter your answers as fractions from smallest to largest.

Answer 1

Combining terms gives

sqrt(2) tan x = sqrt(2)

tan x = 1

x = pi/4 and 5 pi/4

Note that the tan function is y/x or sin x/cos x. Viewing the unit circle, it will be positive in quadrants I and III, but negative in II and IV. There will be infinitely many solutions of the equation; positive or negative integer multiples of 2 pi added to the above solutions would work. However, the problem specifies that x is between 0 and 2 pi, inclusive. So the only 2 correct answers to this problem are pi/4 and 5 pi/4.

Answer 2

4√2 tan x - √2 = 3√2 tan x

√2 (4 tan x - 1) = √2 (3 tan x)

4 tan x - 1 = 3 tan x

tan x - 1 =

tan x = 1

π/4, 5π/4