Solving a Trigonometric Equation

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Discussion Overview

The discussion revolves around solving a trigonometric equation as part of a calculus problem, specifically focusing on the equation 4cos²{x} = sec²{x}/4. Participants explore the process of solving for cos{x} and determining the limits for integration, while also discussing the implications of the solutions within a specific domain.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant expresses uncertainty about solving the equation and seeks help, indicating a need for clarification on the process.
  • Another participant points out the relationship between cos(x) and sec(x), suggesting a foundational understanding of trigonometric identities is necessary.
  • A participant provides a list of trigonometric identities, leading to a transformation of the original equation into a polynomial form.
  • Some participants propose specific solutions for x, such as x = π/3 and x = 2π/3, while noting that there are infinitely many solutions depending on the domain.
  • Concerns are raised about the appropriateness of receiving help on graded homework, with some participants emphasizing the importance of guidance rather than direct answers.
  • A suggestion is made to provide the full context of the problem for more effective assistance.

Areas of Agreement / Disagreement

Participants generally agree on the need for foundational knowledge of trigonometric identities and the process of solving the equation. However, there is no consensus on the complete solution due to the lack of full problem context and the implications of the domain on the solutions.

Contextual Notes

The discussion highlights the importance of understanding the domain of the solutions and the potential for multiple solutions based on that domain. There is also an acknowledgment of the limitations regarding assistance with graded homework.

alane1994
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I have this as part of a calculus problem. I guess I am a little rusty on this. Any help would be appreciated.

4\cos^2{x}=\frac{\sec^2{x}}{4}

I believe that I need to solve for cosx. And then determine what cosx is in radians. That should be the lower limit of the integration. I know how it should work... it is just the process...
 
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re: Solving a Trignometric Equation

Since you brought up the rust, perhaps we can start with that.

Do you know that cos(x) = 1/sec(x)?
 
re: Solving a Trignometric Equation

I know most of the basic ones...
 
re: Solving a Trignometric Equation

Also, I suppose I should give some background on what I am doing. I am finding the area between curves. The trig functions on either side of the = are the two curves.
 
re: Solving a Trignometric Equation

Well, the full list would be:

$$\begin{cases}
\sec x = \frac{1}{\cos x}, \\
\csc x = \frac{1}{\sin x}, \\
\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}.
\end{cases}$$

Therefore, $4 \cos^2 x = \frac{\sec^2 x}{4} = \frac{1}{4} \frac{1}{\cos^2 x}$, leading to $16 \cos^4 x = 1$.

Try to go from here. :D
 
re: Solving a Trignometric Equation

would it be
x= \frac{\pi}{3},\frac{2\pi}{3}
 
re: Solving a Trignometric Equation

alane1994 said:
would it be
x= \frac{\pi}{3},\frac{2\pi}{3}

Those are both solutions yes, but there are infinitely more depending on the domain in question. If the domain is $[0,\pi]$ then those are the only two solutions.
 
re: Solving a Trignometric Equation

I got the answer wrong...
 
re: Solving a Trignometric Equation

alane1994 said:
I got the answer wrong...

Well Mathematica agrees with that answer on that domain. You mentioned that this wasn't the whole problem so without further info I don't know what to tell you.

Also, keep in mind that we can't help you with homework that is graded unless your professor is ok with you receiving any guidance so be careful with this.
 
  • #10
re: Solving a Trignometric Equation

It is indeed graded, but he encourages guidance. Just as long as people are helping you get the answers, not giving them to you. We have an academic center with professors who's entire job consists of helping students with homework questions.
 
  • #11
re: Solving a Trignometric Equation

With Jameson's comment in mind, perhaps this could be solved by posing the question in full. (Nod)
 
  • #12
re: Solving a Trignometric Equation

OK.
I will have to do this in the Calculus section.
 

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