A problem in finding the General Solution of a Trigonometric Equation

In summary, the two solutions are x=(2n+1) \frac{\pi}{2} and x=(2k+1) \frac{\pi}{2} where x is an integer.
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Wrichik Basu
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Homework Statement

:[/B]

Find the general solution of the Trigonometric equation $$\sin {3x}+\sin {x}=\cos {6x}+\cos {4x} $$

Answers given are: ##(2n+1)\frac {\pi}{2}##, ##(4n+1)\frac {\pi}{14}## and ##(4n-1)\frac {\pi}{6}##.

Homework Equations

:[/B]

Equations that may be used:

20170519_023122.png


The Attempt at a Solution

:[/B]

Please see the attached pic:

1495140875722-1985572902.jpg


The answer from Case 1 is correct, but I can't find my mistake in the answers from the two sub-cases of case 2.
 
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  • #2
Wrichik Basu said:

Homework Statement

:[/B]

Find the general solution of the Trigonometric equation $$\sin {3x}+\sin {x}=\cos {6x}+\cos {4x} $$

Answers given are: ##(2n+1)\frac {\pi}{2}##, ##(4n+1)\frac {\pi}{14}## and ##(4n-1)\frac {\pi}{6}##.

Homework Equations

:[/B]

Equations that may be used:

View attachment 203761

The Attempt at a Solution

:[/B]

Please see the attached pic:

View attachment 203760

The answer from Case 1 is correct, but I can't find my mistake in the answers from the two sub-cases of case 2.
Your mistake is in the trigonometric identity ## cosA-cosB=-2sin((A+B)/2)sin((A-B)/2) ##. You didn't divide the terms (A+B and A-B) by 2 in both cases.
 
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  • #3
Charles Link said:
Your mistake is in the trigonometric identity ## cosA-cosB=-2sin((A+B)/2)sin((A-B)/2) ##. You didn't divide the terms (A+B and A-B) by 2 in both cases.
Got it. Thanks a lot.
 
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  • #4
@Wrichik Basu This is an extra detail, but it may interest you that I think the solution ## x=(2n+1) \frac{\pi}{2} ## for all integers ## n ## is actually all included in the other two solutions. The reason is that ## x=(2n+1) \frac{\pi}{2} ## is also always a solution of ## cos(5x)=sin(2x) ##. (A complete expansion of ## cos(5x) ## and ## sin(2x) ##will generate a ## cos(x) ## factor on both sides of the equation.) ## \\ ## You can write ## x= (2k+1) \frac{\pi}{2}=(4n+1) \frac{\pi}{14} ## and if ## k ## is odd, for any ## k ## you can find an integer ## n ##. You can also write ## x=(2k+1) \frac{\pi}{2}=(4m-1) \frac{\pi}{6} ## and if ## k ## is even, for any ## k ## you can find an integer ## m ##. Thereby, the last two solutions completely overlap the ## x=(2n+1) \frac{\pi}{2} ## solution. ## \\ ## Editing... The other two solutions are completely independent of each other=a little algebra shows there is no "x" that is the same in both of them.
 
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Related to A problem in finding the General Solution of a Trigonometric Equation

1. What is a general solution of a trigonometric equation?

A general solution of a trigonometric equation is a solution that is valid for all possible values of the variables involved. In other words, it is a solution that satisfies the equation for any possible input values.

2. Why is finding the general solution of a trigonometric equation important?

Finding the general solution of a trigonometric equation is important because it allows us to solve the equation for any possible input values, rather than just a specific set of values. This provides a more comprehensive understanding of the equation and its solutions.

3. What are some common techniques for finding the general solution of a trigonometric equation?

Some common techniques for finding the general solution of a trigonometric equation include using trigonometric identities, factoring, substitution, and using the unit circle. Additionally, graphing the equation can also help identify possible solutions.

4. Can a trigonometric equation have more than one general solution?

Yes, a trigonometric equation can have more than one general solution. This is because trigonometric functions are periodic, meaning they repeat after a certain interval. Therefore, the equation may have multiple solutions within a given interval.

5. How can I check if a solution is a general solution of a trigonometric equation?

To check if a solution is a general solution of a trigonometric equation, you can substitute the solution into the original equation and see if it satisfies the equation for all possible input values. If it does, then it is a general solution. Additionally, you can also check if the solution satisfies any restrictions or conditions given in the problem.

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