Even & Odd Trigonometric Functions

[nycmathguy]

Section 4.5

Ron Larson said the following:

". . . the sine curve is symmetric with respect to
the origin, whereas the cosine curve is symmetric with respect to the y-axis. These properties of symmetry follow from the fact that the sine function is odd and the cosine function is even."

I don't understand why the sine function is odd based on the fact that it is symmetric with respect to the origin.

I don't understand why the cosine function is even based on the fact that it is symmetric with respect to the y-axis aka the x = 0 line.

Explain.

 

[MathLover1]

Sine Function : f(x) = sin (x)
Graph

Domain: all real numbers
Range: [-1 , 1]
Period = 2pi
x intercepts: x = k pi , where k is an integer.
y intercepts: y = 0
maximum points: (pi/2 + 2 k pi , 1) , where k is an integer.
minimum points: (3pi/2 + 2 k pi , -1) , where k is an integer.
symmetry: since sin(-x) = - sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0 , 0)
intervals of increase/decrease: over one period and from 0 to 2pi, sin (x) is increasing on the intervals (0 , pi/2) and (3pi/2 , 2pi), and decreasing on the interval (pi/2 , 3pi/2).


Cosine Function : f(x) = cos (x)
Graph

Domain: all real numbers
Range: [-1 , 1]
Period = 2pi
x intercepts: x = pi/2 + k pi , where k is an integer.
y intercepts: y = 1
maximum points: (2 k pi , 1) , where k is an integer.
minimum points: (pi + 2 k pi , -1) , where k is an integer.
symmetry: since cos(-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to the y axis.
intervals of increase/decrease: over one period and from 0 to 2pi, cos (x) is decreasing on (0 , pi) increasing on (pi , 2pi).

 

[nycmathguy]

Sine Function : f(x) = sin (x)
Graph

Domain: all real numbers
Range: [-1 , 1]
Period = 2pi
x intercepts: x = k pi , where k is an integer.
y intercepts: y = 0
maximum points: (pi/2 + 2 k pi , 1) , where k is an integer.
minimum points: (3pi/2 + 2 k pi , -1) , where k is an integer.
symmetry: since sin(-x) = - sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0 , 0)
intervals of increase/decrease: over one period and from 0 to 2pi, sin (x) is increasing on the intervals (0 , pi/2) and (3pi/2 , 2pi), and decreasing on the interval (pi/2 , 3pi/2).


Cosine Function : f(x) = cos (x)
Graph

Domain: all real numbers
Range: [-1 , 1]
Period = 2pi
x intercepts: x = pi/2 + k pi , where k is an integer.
y intercepts: y = 1
maximum points: (2 k pi , 1) , where k is an integer.
minimum points: (pi + 2 k pi , -1) , where k is an integer.
symmetry: since cos(-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to the y axis.
intervals of increase/decrease: over one period and from 0 to 2pi, cos (x) is decreasing on (0 , pi) increasing on (pi , 2pi).

Perfect summary. Thanks.