a To decide if a function is even or odd, we replace x with - x and simplify. If we end up with the original function, then the function is even.
A function is even if f(- x)= f(x).
If we end up with the negative of the original function, then the function is odd.
A function is odd if f(- x)= - f(x).
Keeping this information in mind, we want to decide if the trigonometric functions are even or odd.
Sine, Cosine, and Tangent Functions
Recall the Negative Angle Identities.
sin(- θ) &= - sin(θ) [2ex]
cos(- θ) &= cos(θ) [2ex]
tan (- θ) &= - tan (θ)
These identities show that the cosine function is even while the sine and tangent functions are odd. We will now decide if other trigonometric functions are even or odd using these identities.
We obtained the negative of csc θ. Therefore, cosecant function is also odd.
b Instead of trying to think about all possible functions, let's consider just one new function.
f(x)=sin x - cos x
Let's check if this function is even or odd.
Is the Function Even?
A function is even if f(- x)=f(x). To decide if our function is even, we will express the function f( - x) by substituting - x for x. Keep in mind that sine is an odd function, so sin (- x) equals - sin x. Since cosine is an even function, cos (- x) equals cos x.
f( - x)=sin ( - x) - cos ( - x)
sin(- θ)=- sin(θ)
f(- x)=- sin x- cos (- x)
cos(- θ)=cos(θ)
f(- x)=- sin x - cos x
Note that f(- x) is not equal to f(x).
sin x - cos x ≠- sin x - cos x
⇓
f(x) ≠f(- x)
Therefore, the function f(x) is not even.
Is the Function Odd?
A function is odd if f(- x)=- f(x). We have already written an expression for the function f(- x).
f(- x)=- sin x - cos x
We will now express - f(x) by multiplying both sides of the function f(x) by - 1.
Notice that - f(x) is not equal to f(- x).
- sin x - cos x ≠- sin x + cos x
⇓
f(- x) ≠- f(x)
Therefore, the function f(x) is not odd.
Conclusion
We found that f(- x)≠f(x) and f(- x)≠- f(x). Therefore, the function is neither even nor odd. Hence, we cannot say that all functions are even or odd.
