Find all solutions of the equation tan^5 x - 9tan x =0.
The answer is Ak\pi where k is any integer, the constant A=???
To solve the equation tan^5(x) - 9tan(x) = 0, we need to find the values of x that satisfy this equation.
Step 1: Factor out the common term "tan(x)" from both terms:
tan(x) * (tan^4(x) - 9) = 0
Step 2: Set each factor equal to zero and solve for x:
tan(x) = 0 or tan^4(x) - 9 = 0
Step 3: Solve the equation tan(x) = 0:
The solutions for tan(x) = 0 occur when x is an integer multiple of π. Therefore, x = nπ where n is an integer.
Step 4: Solve the equation tan^4(x) - 9 = 0:
Rearrange the equation to isolate the term with the power of 4:
tan^4(x) = 9
Take the square root of both sides:
tan^2(x) = ±√9
Simplify:
tan^2(x) = ±3
Take the square root again:
tan(x) = ±√3
Step 5: Solve the equation tan(x) = ±√3:
The solutions for tan(x) = ±√3 can be found using the inverse tangent function (tan^(-1)):
x = tan^(-1)(√3) + kπ or x = tan^(-1)(-√3) + kπ
where k is any integer.
Therefore, the solutions for the equation tan^5(x) - 9tan(x) = 0 are:
x = nπ, for all integer values of n, and
x = tan^(-1)(√3) + kπ or x = tan^(-1)(-√3) + kπ, where k is any integer.
The constant A is not required to find all the solutions of the equation tan^5(x) - 9tan(x) = 0. It represents a generic constant that can be multiplied with the solutions to give a different representation of the answer.