Solve Trig Inequalities: Find x in [0,2pi] for 2cosx+1≤0

In summary, to find all values of x in the interval [0, 2pi] that satisfy the equation 2cosx + 1 less than or equal to 0, we can first find the roots at x = 2pi/3 and x = 4pi/3. Then, by dividing the interval into three parts and testing for a test point in each interval, we find that the desired answer is (2pi/3, 4pi/3]. If the inequality was only less than (<) instead of less than or equal to (<=), then the interval would be (2pi/3, 4pi/3).
  • #1
EL ALEM
26
0

Homework Statement


find all values of x in the interval [0, 2pi] that satisfy the equation 2cosx + 1 less than or equal to 0


Homework Equations


None.

The Attempt at a Solution


for when 2cosx+1=0
cosx=-1/2
x= 2pi/3, 4pi/3

but what about the values for when 2cosx + 1 is less then 0? How to i fins those?
 
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  • #2
Since you have the roots - why don't you pick values between [0, 2pi/3), (2pi/3, 4pi/3), and (4pi/3, 2pi] to check? One or more of these intervals will give you your desired answer.
 
  • #3
Ok so i found 2cosx + 1 < 0 at this interval (2pi/3, 4pi/3) , so would that be my answer?
 
  • #4
If that is the only interval, then indeed it is! In these types of problems, you can always divide the set into intervals and find test points...
 
  • #5
Thanks a bunch, one more question, if it was only less than instead of less than or equal to how would we divide the set into intervals?
 
  • #6
Oh, well, technically speaking, if your inequality was [itex] \leq [/itex], then you should include the endpoints in the interval. If your inequality was [itex] < [/itex], then you should exclude the endpoints. So what you're really working with is [2pi/3, 4pi/3].
 
  • #7
Ok thanks.
 
  • #8
Ok so i found 2cosx + 1 < 0 at this interval (2pi/3, 4pi/3) , so would that be my answer?
 

Related to Solve Trig Inequalities: Find x in [0,2pi] for 2cosx+1≤0

What is a trig inequality?

A trig inequality is an inequality that involves trigonometric functions, such as sine, cosine, and tangent. These inequalities typically involve finding the values of an unknown variable that satisfy the inequality.

How do I solve trig inequalities?

To solve a trig inequality, you first need to isolate the trigonometric function on one side of the inequality. Then, you can use algebraic techniques and the properties of trigonometric functions to solve for the unknown variable.

What does [0,2pi] mean?

[0,2pi] is a notation used to indicate the range of values for the variable x. In this case, it means that the values of x must be between 0 and 2pi, inclusive.

What is the significance of the inequality 2cosx+1≤0?

The inequality 2cosx+1≤0 represents a range of values for x that satisfy the given equation. In this case, it means that the values of x must be between 0 and 2pi, and the cosine of x must be less than or equal to -1/2.

What is the solution to the inequality 2cosx+1≤0?

The solution to this inequality is x = π/3, 2π/3, 4π/3, and 5π/3. These values satisfy the inequality and fall within the specified range of [0,2pi].

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