Domain: all real numbers except k pi, k is an integer.
Range: all real numbers
Period = pi
X intercepts: x = pi /2 + k pi , where k is an integer.
Symmetry: since cot(-x) = - cot(x) then cot (x) is an odd function and its graph is symmetric with respect the origin.
Intervals of increase/decrease: over one period and from 0 to pi, cot (x) is decreasing.
Vertical asymptotes: x = k pi, where k is an integer.
Secant
Domain: all real numbers except pi/2 + k pi, n is an integer.
Range: (-infinity , -1] U [1 , +infinity)
Period = 2 pi
Y intercepts: y = 1
Symmetry: since sec(-x) = sec (x) then sec (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 2 pi, sec (x) is increasing on (0 , pi/2) U (pi/2 , pi) and decreasing on (pi , 3pi/2) U (3pi/2 , 2pi).
Vertical asymptotes: x = pi/2 + k pi, where k is an integer.
Cosecant
Domain: all real numbers except k pi, k is an integer.
Range: (-infinity , -1] U [1 , +infinity)
Period = 2pi
Symmetry: since csc(-x) = - csc(x) then csc (x) is an odd function and its graph is symmetric with respect the origin.
Intervals of increase/decrease: over one period and from 0 to 2pi, csc (x) is decreasing on (0 , pi/2) U (3pi/2 , 2pi) and increasing on (pi/2 , pi) U (pi / 3pi/2).
Vertical asymptotes: x = k pi, where k is an integer.
Tangent
Domain: all real numbers except pi/2 + k pi, k is an integer.
Range: all real numbers
Period = pi
X intercepts: x = k pi , where k is an integer.
Y intercepts: y = 0
Symmetry: since tan(-x) = - tan(x) then tan (x) is an odd function and its graph is symmetric with respect the origin.
Intervals of increase/decrease: over one period and from -pi/2 to pi/2, tan (x) is increasing.
Vertical asymptotes: x = pi/2 + k pi, where k is an integer.
Cosine
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).
Sine
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 origon (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).
