Why are Cosine and Secant Even Functions?

In summary, the cosine and secant functions are even because their graphs are symmetric at the y-axis, meaning that what appears on the left side appears again on the right side. This is different from the other trigonometric functions whose graphs do not exhibit this behavior. Additionally, the definition of cosine as the x-coordinate of a point on the unit circle explains why cos(-t) = cos(t) and sec(-t) = sec(t). The concept can also be understood through the use of power series, which shows that cosine is made up of only even terms. However, this may not be helpful if the person asking the question is not familiar with power series.
  • #1
CrossFit415
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Homework Statement



Why is Cosin and Secant even? Cos (-t) = cos t, Sec (-t) = sec t
Why don't they equal - sec t instead like the rest of the functions? Thanks


Homework Equations





The Attempt at a Solution

 
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  • #2
You could look at their graphs. Even functions have graphs that are symmetric at the y-axis -- what appears on the left side of the y-axis appears again as a "mirror" on the right side of the y-axis. The graphs of the rest of the trig functions do not exhibit this behavior.
 
  • #3
Hmm I somewhat kind of get it now. Thanks a lot!
 
  • #4
How, exactly, are you defining "cosine". Probably you are using something like "Given a number t, measure a distance t around the circumference of the unit circle, starting at (1, 0). cos(t) is the x coordinate of the ending point." From that it should be clear that if t> 0 takes you to the point (x, y), -t takes you to (x, -y). x= cos(t)= cos(-t), y= sin(t), -y= sin(-t).
 
  • #5
The reason why cosine is an even functions is because if you expand cosine via power series you'll get polynomials that are only even: 1-X2/2!+X4/4!-X6/6!+X8/8!...
 
  • #6
romsofia said:
The reason why cosine is an even functions is because if you expand cosine via power series you'll get polynomials that are only even: 1-X2/2!+X4/4!-X6/6!+X8/8!...

I don't think this is helpful to the OP, because, unless I'm mistaken, he/she hasn't seen power series yet.
 
  • #7
eumyang said:
I don't think this is helpful to the OP, because, unless I'm mistaken, he/she hasn't seen power series yet.

I thought about that before posting, but I'm sure it'll be useful to see why it's really an even function.
 
  • #8
In fact, it is possible to define cos(x) in terms of its Taylor series. For such a definition, romsofia's response is perfect. But we don't know what kind of response is appropriate until we know exactly how the OP is defining cosine and secant.
 

What is the definition of Cosine and Secant?

Cosine and Secant are trigonometric functions commonly used in mathematics and physics. Cosine (cos) is the ratio of the adjacent side to the hypotenuse of a right triangle, while Secant (sec) is the inverse of the cosine function.

Why is Cosine and Secant even?

Cosine and Secant are even functions because they exhibit symmetry about the y-axis. This means that the value of the function at any point on the positive x-axis is equal to the value at the same point on the negative x-axis.

How is the evenness of Cosine and Secant useful?

The evenness of Cosine and Secant allows for simplification and manipulation of trigonometric equations. For example, if the variable x appears in an even power in an equation, the evenness of cosine and secant allows us to replace it with the absolute value of x.

Can the evenness of Cosine and Secant be proved?

Yes, the evenness of Cosine and Secant can be proved using the unit circle or by using the properties of even functions, such as the property that f(-x) = f(x).

Are there any other trigonometric functions that are even?

Yes, there are two other trigonometric functions that are even - the Even Cosine (cosec) and Even Tangent (cot). These functions exhibit symmetry about the y-axis just like cosine and secant.

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