Find the solutions to the given equation on the interval [0,2π)?
cot^3x−cotx=0
I don't understand this...angles to recognize etc..
please explain..thank you
cot^3x−cotx=0
cotx(cot^2 x - 1) = 0
cotx(cotx+1)(cotx-1) = 0
so cotx = 0 , cotx = -1, cotx = 1
consider the last two:
if cotx = -1, cotx = 1
tanx = -1, tanx = 1
which means that x could be in any of the 4 quadrants
you must know that tan π/4 = 1 , (tan 45° =1)
so x = π/4 , 3π/4, 5π/4, and 7π/4
let's look at the cotx = 0
mathematically, then tanx = 1/0, which of course is undefined
which means that the tangent curve must have an asymptote at that value of x
looking at the tangent curve , we can see that at x = π/2 , or 90°, the tangent is undefined
The period of the tangent curve is π, so we have another solution at
π/2 + π = 3π/2
so you have 6 solutions for your equation:
x = π/2, 3π/2, π/4 , 3π/4, 5π/4, 7π/4
You should memorize the basic trig ratios for angles of
30°, 45°, 60° and 90° from the 2 simple right-angled triangles with sides
of 1-1-√2, and 3-4-5
Thank you so,so much for that explanation!
To find the solutions to the equation cot^3x - cotx = 0 on the interval [0,2π), we need to solve for x.
First, let's simplify the equation. By factoring out a common factor of cotx, we get:
cotx(cot^2x - 1) = 0
Now, we can use the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be zero. So we set each factor equal to zero and solve the resulting equations.
Setting cotx = 0, we can rearrange the equation to solve for x:
cotx = 0
1/tanx = 0
tanx = 1/0 (undefined)
Now let's solve the equation cot^2x - 1 = 0:
cot^2x = 1
1/tan^2x = 1
tan^2x = 1
tanx = ± √1
tanx = ±1
Now, we need to find the values of x on the interval [0, 2π) for which these equations are true.
For the equation cotx = 0:
Since cotx is undefined when tanx = 0, there are no solutions on the interval [0, 2π).
For the equation tanx = 1:
The solutions on the interval [0, 2π) are x = π/4 and x = 5π/4.
For the equation tanx = -1:
The solutions on the interval [0, 2π) are x = 3π/4 and x = 7π/4.
Therefore, the solutions to the equation cot^3x - cotx = 0 on the interval [0,2π) are x = π/4, 3π/4, 5π/4, and 7π/4.