Which of the six trigonometric functions are even functions?

Answer 1

A function f(x) is even if:

f(x) = f(-x)

In layman's terms this property simply means that any real number in the domain and it's opposite will yield the same function value in the range.

To simplify this down even further, an even function, when graphed will appear to be symetric about the y-axis (assuming that you use the standard Cartesian coordinate plane).

In the case of trig functions, you would have to test whether the even function property holds true for each. We will the test points π, π/2, or π/4. NOTE: The # signs are present next to the functions that are even:

1. Sine: f(x) = sin(x)

-> sin(π/2) = 1, but sin(-π/2) = -1. Since 1 does not equal -1, sine is NOT an even function.

2. #Cosine: f(x) = cos(x)

-> cos(π) = -1 = cos(-π). Since both are equal, cosine IS an even function.

3. Tangent: f(x) = tan(x)

-> tan(π/4) = 1, but tan(-π/4) = -1. Therefore, tangent is NOT an even function.

4. Cosecant: f(x) = csc(x)

-> csc(π/2) = 1, but csc(-π/2) = -1. Therefore, cosecant is NOT an even function.

5. #Secant: f(x) = sec(x)

-> sec(π) = -1 = sec(-π). Since the secant function has asymptotes, it IS an even function provided that x does not equal π(2n+1)/2, where n may be all integers.

6. Cotangent: f(x) = cot(x)

-> cot(π/4) = 1, but cot(-π/2) = -1. Therefore cotangent is NOT even.