Simplifying a cosine + cosine with conjugate denominators

In summary, the conversation discusses an attempted simplification using trigonometric functions and the confusion over the result. The problem does not mention any restrictions on the variable n and it is suggested that the original expression may have come from an incorrectly worked Fourier Series problem.
  • #1
luckyduck
7
0

Homework Statement



-[itex]\frac{1}{2}[/itex][cos([itex]\frac{\pi+\pi n}{\pi+\pi n}[/itex]) + cos([itex]\frac{\pi-\pi n}{\pi-\pi n}[/itex])]

Homework Equations



cos(u)cos(v) = [itex]\frac{1}{2}[/itex] cos(u+v)+cos(u-v)

The Attempt at a Solution


I am attempting to use the above trig function to simplify the first function, but I can't seem to do it properly. Is there another function for when the contents of the cos are equal to its denominator?

Thanks in advance for all your help!
 
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  • #2
luckyduck said:

Homework Statement



-[itex]\frac{1}{2}[/itex][cos([itex]\frac{\pi+\pi n}{\pi+\pi n}[/itex]) + cos([itex]\frac{\pi-\pi n}{\pi-\pi n}[/itex])]

Homework Equations



cos(u)cos(v) = [itex]\frac{1}{2}[/itex] cos(u+v)+cos(u-v)

The Attempt at a Solution


I am attempting to use the above trig function to simplify the first function, but I can't seem to do it properly. Is there another function for when the contents of the cos are equal to its denominator?

Thanks in advance for all your help!
Don't the fractions simplify to 1 for nearly all values of n? BTW, are there any restrictions on n?
 
  • #3
I keep getting weird numbers. Problem doesn't state any restrictions!
 
  • #5
luckyduck said:

Homework Statement



-[itex]\frac{1}{2}[/itex][cos([itex]\frac{\pi+\pi n}{\pi+\pi n}[/itex]) + cos([itex]\frac{\pi-\pi n}{\pi-\pi n}[/itex])]

Mark44 said:
What are you getting?

More to the point, where did that expression come from? I'm guessing almost certainly from an incorrectly worked Fourier Series problem.
 

1. What does it mean to simplify a cosine + cosine with conjugate denominators?

Simplifying a cosine + cosine with conjugate denominators refers to the process of combining two cosine functions with conjugate denominators into one simplified expression. This is typically done to make the expression easier to work with and to eliminate any complex or repeated terms.

2. How do I identify conjugate denominators in a cosine expression?

Conjugate denominators in a cosine expression are pairs of expressions that differ only in the sign between their terms. For example, the expressions (a + b) and (a - b) are conjugate denominators. To identify conjugate denominators, look for expressions with the same terms but with opposite signs.

3. What is the process for simplifying a cosine + cosine with conjugate denominators?

The process for simplifying a cosine + cosine with conjugate denominators involves multiplying the entire expression by the conjugate of the denominator and then using the identity cos^2(x) + sin^2(x) = 1 to eliminate any complex or repeated terms. This results in a simplified expression with a single cosine term.

4. Can I simplify expressions with more than two cosines and conjugate denominators?

Yes, expressions with more than two cosines and conjugate denominators can be simplified using the same process of multiplying by the conjugate of the denominators and applying the cos^2(x) + sin^2(x) = 1 identity. The end result will still be a simplified expression with a single cosine term.

5. Why is simplifying a cosine + cosine with conjugate denominators important in mathematics?

Simplifying a cosine + cosine with conjugate denominators is important in mathematics because it allows for easier manipulation and calculation of expressions involving cosine functions. It also helps to reduce the complexity of an expression and make it more manageable to work with in further calculations or applications.

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