Solving trigonometric inequalities

In summary, the conversation discusses solving a trigonometric inequality and presents two possible solutions. The speaker asks for help and the responder suggests finding the intervals that satisfy both the inequality and the condition on t. The conversation ends with the responder providing a hint on how to solve the problem.
  • #1
LovePhys
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Hello,

I am struggling with solving trigonometric inequalities. For example, solve: [itex]cos(\frac{\pi t}{3}) < \frac{1}{2}[/itex], [itex] 0<t<50 [/itex]
I wonder if one of these solutions is true:
1/ [itex] \frac{\pi}{3} + k2\pi < \frac{\pi t}{3} < \frac{5\pi}{3} + k2\pi, k \in Z [/itex]
2/ [itex] \frac{\pi}{3} + 6k < \frac{\pi t}{3} < \frac{5\pi}{3} + 6k, k \in Z [/itex] (the period of [itex]cos(\frac{\pi t}{3})[/itex] is 6)
I checked both of them and it seemed that the first solution is correct. However, personally, I think both of them are correct:
1/ The first solution: For example, we got the solution [itex] \frac{2\pi}{3}[/itex]. Obviously, it'll repeat with the period of [itex]2\pi[/itex] on the unit circle.
2/ The second solution: If we got one solution, it'll repeat with the period of 6 on the graph of [itex]cos(\frac{\pi t}{3})[/itex].
I have been struggling with this problem for a long time, yet I cannot figure it out.
Hopefully I can be given a little help.
Thanks a bunch everyone!
Huyen Nguyen
 
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  • #2
If you leave them this way, both of them are not correct.

Think of which is the variable you need to find, and the limits on it posed by the problem.
The problem asks to find t.
So, try to put the solutions you wrote in the form ...<t<...
Then, you must find the intervals that satisfy both the trigonometric inequality and the condition on t.

What does this mean?
 
  • #3
@ DriracRules: Thanks for your response. I have been thinking about this problem, yet I haven't figured it out. If you know the answer, can you please help me? Thank you very much.
 
  • #4
First, note that [itex]cos(\pi/3)= 1/2[/itex] and cosine is decreasing between 0 and [itex]\pi[/itex] so, immediately, cos(x)< 1/2 for [itex]\pi/3< x\le \pi[/itex] which, for this problem, gives [itex]\pi/2< \pi t/3\le \pi[/itex]. Solve for t and then use the periodicity of cosine to extend to values of t< 50.
 

1. What are trigonometric inequalities?

Trigonometric inequalities are mathematical statements that involve trigonometric functions, such as sine, cosine, and tangent, and compare them using the symbols <, >, ≤, and ≥.

2. How do I solve trigonometric inequalities?

To solve trigonometric inequalities, you need to use algebraic techniques and trigonometric identities to isolate the variable and determine the intervals where the inequality is true.

3. What are the key steps for solving trigonometric inequalities?

The key steps for solving trigonometric inequalities are: 1) Isolate the variable on one side of the inequality, 2) Use trigonometric identities to simplify the expression, 3) Determine the intervals where the inequality is true, and 4) Check the endpoints of the intervals to ensure the inequality holds true.

4. How do I know which trigonometric identities to use when solving trigonometric inequalities?

When solving trigonometric inequalities, you can use any trigonometric identities that help you simplify the expression and isolate the variable. Some commonly used identities include the Pythagorean identities, double angle identities, and sum and difference identities.

5. Are there any special cases to consider when solving trigonometric inequalities?

Yes, there are a few special cases to consider when solving trigonometric inequalities. These include when the inequality involves a tangent function, when the inequality is in radians instead of degrees, and when the inequality involves multiple trigonometric functions.

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