0
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.
Still curious? Ask our experts.
Chat with our AI personalities
Ezra
Taiga
RossAdd your answer:
Earn +20 pts
What is the difficulties of trigonometric function of an angles?
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
Which trigonometric functions have asymptotes?
tangent, cosecants, secant, cotangent.
Definition of the 6 trigonometric functions?
The six main trigonometric functions are sin(x)=opposite/hypotenuse cos(x)=adjacent/hypotenuse tan(x)=opposite/adjacent csc(x)=hypotenuse/opposite cot(x)=adjacent/opposite sec(x)=hypotenuse/adjacent Where hypotenuse, opposite, and adjacent correspond to the three sides of a right triangle and x corresponds to an angle in that right triangle.
What is unary?
It refers to one.A binary function (binary = 2) takes two numbers as input and gives the result (output) as a single number. Thus, addition is a binary function. Some functions, like squaring or trigonometric functions are examples of unary functions. These have only one input.
What function family has an increasing interval and a decreasing interval?
There are many families of functions or function types that have both increasing and decreasing intervals. One example is the parabolic functions (and functions of even powers), such as f(x)=x^2 or f(x)=x^4. Namely, f(x) = x^n, where n is an element of even natural numbers. If we let f(x) = x^2, then f'(x)=2x, which is < 0 (i.e. f(x) is decreasing) when x<0, and f'(x) > 0 (i.e. f(x) is increasing), when x > 0. Another example are trigonometric functions, such as f(x) = sin(x). Finding the derivative (i.e. f'(x) = cos(x)) and critical points will show this.
