Understanding Odd and Even Functions in Trigonometry

In summary, an odd function is a mathematical function where the output value changes sign when the input value changes sign. Conversely, an even function is a mathematical function where the output value remains the same when the input value changes sign. To determine if a function is odd or even, you can use the symmetry test or check the power of the highest degree term. Examples of odd functions include f(x) = x^3, f(x) = -sin(x), and f(x) = -x, while examples of even functions include f(x) = x^2, f(x) = cos(x), and f(x) = |x|. Odd and even functions are symmetric about different axes, with odd functions being symmetric about the origin and
  • #1
yungman
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Just want to verify

##e^{jx\cos(\theta)}## is EVEN function respect to ##\theta## as ##e^{jx\cos(-\theta)}=e^{jx\cos(\theta)}##

BUT

##e^{jx\sin\theta\;}## is neither ODD nor EVEN because ##\;e^{jx\sin(-\theta)}=e^{-jx\sin(\theta)}\neq\;^+_- e^{jx\sin\theta}##

Thanks
 
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  • #2
True.
 
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  • #3
dirk_mec1 said:
True.

Thanks

Also ##\sin(x\cos\theta)## is an EVEN function as ##\sin[x\cos(-\theta)]=sin(x\cos\theta)##.

Thanks for your help.
 
  • #4
Yes, w.r.t to theta.
 
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  • #5
dirk_mec1 said:
Yes, w.r.t to theta.

Thanks

What is w.r.t?
 
  • #6
w.r.t. = 'with respect to'
 
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1. What is an odd function?

An odd function is a mathematical function where the output value changes sign when the input value changes sign. In other words, for any input value x, the output value f(x) of an odd function will be equal to -f(-x). This results in the graph of an odd function being symmetric about the origin.

2. What is an even function?

An even function is a mathematical function where the output value remains the same when the input value changes sign. In other words, for any input value x, the output value f(x) of an even function will be equal to f(-x). This results in the graph of an even function being symmetric about the y-axis.

3. How do you determine if a function is odd or even?

To determine if a function is odd or even, you can use the symmetry test. Plug in -x for x in the function and see if the output value remains the same (even function) or changes sign (odd function). Another way is to check the power of the highest degree term in the function. If it is an even power, the function is even. If it is an odd power, the function is odd.

4. What are some examples of odd and even functions?

Examples of odd functions include f(x) = x^3, f(x) = -sin(x), and f(x) = -x. Examples of even functions include f(x) = x^2, f(x) = cos(x), and f(x) = |x|.

5. How do odd and even functions relate to symmetry?

Odd and even functions are symmetric about different axes. Odd functions are symmetric about the origin, while even functions are symmetric about the y-axis. This means that if you fold the graph of an odd function in half along the y-axis, the two halves will match up. For even functions, if you fold the graph along the y-axis, the two halves will also match up. This symmetry property helps in understanding the behavior of these functions and their graphs.

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