How do you find all the solutions of the equation in the interval [0,2π)?
3tan^3x=tanx
tan x (3tan^2 x - 1) = 0
tan x = 0 or tan x = +/- 1/sqrt3 are both solutions
There are four angles that have a tangent of +or- 1/sqrt3, and two angles with a tangent of zero in the [0,2 pi) interval
To find all the solutions of the equation within the given interval, follow these steps:
Step 1: Rewrite the equation using trigonometric identities.
In this case, we can use the identity tan^2(x) = sec^2(x) - 1 and tan^3(x) = tan(x) * tan^2(x).
The equation can be rewritten as:
3tan(x) * (sec^2(x) - 1) = tan(x).
Step 2: Simplify the equation.
Expand the equation using the identities:
3sec^2(x)tan(x) - 3tan(x) = tan(x).
Step 3: Bring all terms to one side of the equation.
3sec^2(x)tan(x) - 3tan(x) - tan(x) = 0.
Step 4: Factor out the common term tan(x).
tan(x)(3sec^2(x) - 4) = 0.
Step 5: Set each factor equal to zero and solve for x.
tan(x) = 0:
In the interval [0, 2π), the solutions for tan(x) = 0 are x = 0, π.
3sec^2(x) - 4 = 0:
Rearrange the equation: sec^2(x) = 4/3.
Take the square root of both sides: sec(x) = ±√(4/3).
To find the values of x that satisfy sec(x) = √(4/3), use the inverse cosine function:
x = cos^(-1)(√(4/3)).
Solve for x:
x ≈ 0.841, or x ≈ 5.981.
To find the values of x that satisfy sec(x) = -√(4/3), use the inverse cosine function again:
x = cos^(-1)(-√(4/3)).
Solve for x:
x ≈ 2.301, or x ≈ 4.521.
So, the solutions in the interval [0, 2π) are x = 0, π, 0.841, 2.301, 4.521, and 5.981.