Find all solutions in the interval [0, 2π). 7 tan³x - 21 tan x = 0

Solution:

Given, 7 tan³x - 21 tan x = 0

The interval is [0, 2π]

Let tan x = a

Now the equation becomes,

7a³ - 21a = 0

a(7a² - 21) = 0

Now, a = 0

7a² - 21 = 0

7a² = 21

a² = 21/7

a² = 3

Taking square root,

a = ±√3

So, the value of a is 0, +√3 and -√3.

When tan x = 0

x = tan^-1(0)

x = 0 or π

When tan x = +√3

x = tan^-1(+√3)

x = π/3 or 4π/3

When tan x = -√3

x = tan^-1(-√3)

x = 2π/3 or 5π/3

Therefore, the solutions of 7 tan²x - 21 tan x = 0 are 0, π, π/3, 4π/3, 2π/3, 5π/3.

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Find all solutions in the interval [0, 2π). 7 tan²x - 21 tan x = 0

Summary:

The solutions of 7 tan²x - 21 tan x = 0 in the interval [0, 2π) are 0, π, π/3, 4π/3, 2π/3, 5π/3.