Properties of Trigonometric Functions
The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x) are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points.
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).
Tangent Function : f(x) = tan (x)

Graph
 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.
Cotangent Function : f(x) = cot (x)

Graph
 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 Function : f(x) = sec (x)

Graph
 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 Function : f(x) = csc (x)

Graph
 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.
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trigonometric functions