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3.1.5: Even and Odd Identities

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Functions symmetric with respect to the y-axis or about the origin.

You and your friend are in math class together. You enjoy talking a lot outside of class about all of the interesting topics you cover in class. Lately you've been covering trig functions and the unit circle. As it turns out, trig functions of certain angles are pretty easy to remember. However, you and your friend are wishing there was an easy way to ‘‘shortcut’’ calculations so that if you knew a trig function for an angle you could relate it to the trig function for another angle; in effect giving you more reward for knowing the first trig function.

You're examining some notes and starting writing down trig functions at random. You eventually write down:

cos(π18)

Is there any way that if you knew how to compute this, you'd automatically know the answer for a different angle?

An even function is a function where the value of the function acting on an argument is the same as the value of the function when acting on the negative of the argument. Or, in short:

f(x)=f(x)

So, for example, if f(x) is some function that is even, then f(2) has the same answer as f(2). f(5) has the same answer as f(5), and so on.

In contrast, an odd function is a function where the negative of the function's answer is the same as the function acting on the negative argument. In math terms, this is:

f(x)=f(x)

If a function were negative, then f(2)=f(2), f(5)=f(5), and so on.

Functions are even or odd depending on how the end behavior of the graphical representation looks. For example, y=x2 is considered an even function because the ends of the parabola both point in the same direction and the parabola is symmetric about the y−axis. y=x3 is considered an odd function for the opposite reason. The ends of a cubic function point in opposite directions and therefore the parabola is not symmetric about the y−axis. What about the trig functions? They do not have exponents to give us the even or odd clue (when the degree is even, a function is even, when the degree is odd, a function is odd).

 Even Function y=(x)2=x2 Odd Function y=(x)3=x3

Let’s consider sine. Start with sin(x). Will it equal sinx or sinx? Plug in a couple of values to see.

sin(30)=sin330=12=sin30sin(135)=sin225=22=sin135

From this we see that sine is odd. Therefore, sin(x)=sinx, for any value of x. For cosine, we will plug in a couple of values to determine if it’s even or odd.

cos(30)=cos330=32=cos30cos(135)=cos225=22=cos135

This tells us that the cosine is even. Therefore, cos(x)=cosx, for any value of x. The other four trigonometric functions are as follows:

tan(x)=tanxcsc(x)=cscxsec(x)=secxcot(x)=cotx

Notice that cosecant is odd like sine and secant is even like cosine.

Finding Even and Odd Identities

1. Find sinx

If cos(x)=34 and tan(x)=73, find sinx.

We know that sine is odd. Cosine is even, so cosx=34. Tangent is odd, so tanx=73. Therefore, sine is positive and sinx=74.

2. Find sin(-x)

If sin(x)=.25, find sin(x)

Since sine is an odd function, sin(θ)=sin(θ).

Therefore, sin(x)=sin(x)=.25

3. Find cos(-x)

If cos(x)=.75, find cos(x)

Since cosine is an even function, cos(x)=cos(x).

Therefore, cos(x)=.75

Example 3.1.5.1

Earlier, you were asked to compute cos(π18).

Solution

Since you now know that cosine is an even function, you get to know the cosine of the negative of an angle automatically if you know the cosine of the positive of the angle.

Therefore, since cos(π18)=.9848, you automatically know that cos(π18)=cos(17π18)=.9848.

Example 3.1.5.2

What two angles have a value for cosine of 32?

Solution

On the unit circle, the angles 30 and 330 both have 32 as their value for cosine. 330 can be rewritten as 30

Example 3.1.5.3

If cosθ=32, find sec(θ)

Solution

There are 2 ways to think about this problem. Since cosθ=cosθ, you could say sec(θ)=1cos(θ)=1cos(θ) Or you could leave the cosine function the way it is and say that sec(θ)=sec(θ)=1cosθ. But either way, the answer is 23

Example 3.1.5.4

If cotθ=3 find cotθ

Solution

Since cot(θ)=cot(θ), if cotθ=3 then cot(θ)=3. Therefore, cot(θ)=3.

Review

Identify whether each function is even or odd.

  1. y=sin(x)
  2. y=cos(x)
  3. y=cot(x)
  4. y=x4
  5. y=x
  6. If sin(x)=.3, what is sin(x)?
  7. If cos(x)=.5, what is cos(x)?
  8. If tan(x)=.1, what is tan(x)?
  9. If cot(x)=.3, what is cot(x)?
  10. If csc(x)=.3, what is csc(x)?
  11. If sec(x)=2, what is sec(x)?
  12. If sin(x)=.2, what is sin(x)?
  13. If cos(x)=.25, what is sec(x)?
  14. If csc(x)=4, what is sin(x)?
  15. If tan(x)=.2, what is cot(x)?
  16. If sin(x)=.5 and cos(x)=32, what is cot(x)?
  17. If cos(x)=.5 and sin(x)=32, what is tan(x)?
  18. If cos(x)=22 and tan(x)=1, what is sin(x)?

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.

Vocabulary

Term Definition
Even Function An even function is a function with a graph that is symmetric with respect to the y-axis and has the property that f(x)=f(x).
Odd Function An odd function is a function with the property that f(x)=f(x). Odd functions have rotational symmetry about the origin.

Additional Resources

Video: Even and Odd Trigonometric Identities

Practice: Even and Odd Identities


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