Trigonometric functions are mathematical concepts that describe the relationships between the angles and sides of triangles. The primary functions include tangent, sine, cosecant, and cosine, each with distinct properties and behaviors.
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).
