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 find all the solutions of the equation tan^5 x - 9tan x = 0, we can rearrange the equation to get:
tan^5 x = 9tan x
Now, we can set tan x equal to zero or take the fifth root of both sides:
tan x = 0 or tan x = (9tan x)^(1/5)
Let's solve each case separately:
Case 1: tan x = 0
To find the solutions for this case, we need to know the values of x where the tangent function equals zero. The tangent function is equal to zero at multiples of π (pi), so we can write:
x = kπ
where k is any integer.
Case 2: tan x = (9tan x)^(1/5)
To find the solutions for this case, we need to simplify the equation:
(tan^4 x)^(1/5) = 9^(1/5) * (tan x)^(1/5)
Taking the fifth power of both sides gives us:
tan^4 x = 9 * tan x
Now, we can substitute u = tan x to simplify the equation further:
u^4 = 9u
Rearranging the equation gives us:
u^4 - 9u = 0
Factoring out u, we get:
u(u^3 - 9) = 0
Now, we have two possibilities:
a) u = 0
b) u^3 - 9 = 0
For case a), u = 0 implies tan x = 0. We already found the solutions for this case in Case 1, which are x = kπ.
For case b), to find the solutions for u^3 - 9 = 0, we solve the equation:
u^3 - 9 = 0
Using the difference of cubes formula, we can factorize the equation:
(u - ∛9)(u^2 + ∛9u + (∛9)^2) = 0
Now, we have two possibilities:
i) u - ∛9 = 0
ii) u^2 + ∛9u + (∛9)^2 = 0
For i), u - ∛9 = 0 implies u = ∛9. Plugging this back into the substitution we made before (u = tan x), we get:
tan x = ∛9
To find the values of x for this case, we can use the inverse tangent function:
x = arctan(∛9) = A
(where A is a constant)
For ii), solving the quadratic equation u^2 + ∛9u + (∛9)^2 = 0 gives us:
u = (-∛9 ± √(9 - 4(∛9)^2)) / 2
Simplifying further, we find:
u = (-∛9 ± √(9 - 36∛9)) / 2
However, this equation doesn't have real solutions since the term inside the square root is negative. Therefore, there are no solutions for this case.
In summary, the solutions to the equation tan^5 x - 9tan x = 0 are:
1) x = kπ (where k is any integer)
2) x = A (where A is a constant)