Which of the following are solutions to the given equation on the interval [0,2π)?
cot^3x−cotx=0
Select all that are true. If there are none that are true, do no select any before submitting.
π/6
0
5π/4
3π/4
π
π/3
11π/6
7π/4
π/4
π/2
help please
cot x (cot^2 x - 1) = 0
when does cot x = 0 ?
cot x = cos x /sin x
works at x = pi/2 and x= 3 pi/2
when does cot^2 x = +1
cos x = sin x at x = pi/4 and at x = 5 pi/4
Ok..so what is the answer? Confused
To find the solutions to the given equation cot^3x - cotx = 0 on the interval [0, 2π), we can follow these steps:
Step 1: Rewrite the equation using the reciprocal identity for cotangent:
1 - tan^2x - (1/tanx) = 0
Step 2: Multiply through by tanx to eliminate fractions:
tanx - tan^3x - 1 = 0
Step 3: Rearrange the equation to a cubic equation form:
tan^3x - tanx + 1 = 0
Now that we have the equation in a cubic form, we can find the solutions:
Step 4: Use numerical or algebraic methods to solve the cubic equation. In this case, we can use trial and error to find the solutions.
Given the options, let's test each angle to see if it satisfies the equation:
1. For π/6 ≈ 0.5236:
Substitute tan(π/6) = (√3)/3 into the equation:
( (√3)/3 )³ - (√3)/3 + 1 ≈ 0.192 <≠ 0
2. For 0:
Substitute tan(0) = 0 into the equation:
0 - 0 + 1 = 1 ≠ 0
3. For 5π/4 ≈ 3.927:
Substitute tan(5π/4) = 1 into the equation:
1 - 1 + 1 = 1 ≠ 0
4. For 3π/4 ≈ 2.356:
Substitute tan(3π/4) = -1 into the equation:
(-1)³ - (-1) + 1 = -1 ≠ 0
5. For π:
Substitute tan(π) = 0 into the equation:
0 - 0 + 1 = 1 ≠ 0
6. For π/3 ≈ 1.047:
Substitute tan(π/3) = √3 into the equation:
( √3 )³ - √3 + 1 ≈ 1 ≠ 0
7. For 11π/6 ≈ 5.759:
Substitute tan(11π/6) = -(√3)/3 into the equation:
( -(√3)/3 )³ - (-(√3)/3) + 1 ≈ 0.192 <≠ 0
8. For 7π/4 ≈ 5.498:
Substitute tan(7π/4) = -1 into the equation:
(-1)³ - (-1) + 1 = -1 ≠ 0
9. For π/4 ≈ 0.785:
Substitute tan(π/4) = 1 into the equation:
1 - 1 + 1 = 1 ≠ 0
10. For π/2 ≈ 1.571:
Substitute tan(π/2) = ∞ into the equation:
(∞)³ - ∞ + 1 = ∞ ≠ 0
Based on our calculations, the only angle that satisfies the equation is π/6. Thus, the solution to the equation cot^3x - cotx = 0 on the interval [0, 2π) is π/6.