1. How can I tell if a trig function is even, odd or neither using its graph?
if the graph is symmetric with respect to the origin , a function is odd
if the graph is symmetric to the y- axis, a function is an even function
The majority of functions are neither odd nor even, however, sine and tangent are odd functions and cosine is an even function.
All functions, including trig functions, can be described as being even, odd, or neither.
A
function is odd if and only if
f(-x) = - f(x) and is symmetric with respect to the origin.
A
function is even if and only if
f(-x) = f(x) and is symmetric to the y axis.
It is helpful to know if a function is odd or even when you are trying to simplify an expression when the variable inside the trigonometric function is negative.
sin( -x ) = - sin (x) ->a function is odd
cos ( -x ) = cos (x)->a function is even
tan ( -x ) = - tan( x)->a function is odd
cot(-x) = -cot(x) ->a function is odd
sec (-x ) = sec (x)->a function is even
csc ( -x ) = - csc (x)->a function is odd
2.
prove that tangent is an odd function
A function is odd if:
f(-x)=-f(x)
tan(-x)=sin(-x)/cos(-x)=-sin(x)/cos(x)=-sin(x)/cos(x)=-tan(x)
more here:
3.
61. f(x)=x+tan(x)
using a graph
View attachment 1064
The graph is symmetric with respect to the origin
therefore it is on odd function.
algebraically:
f(-x) = -x - tan(-x) = -(x - tan(x)) = -f(x)
