Mathematics Graphs & Properties of Trigonometric Functions
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Topics Covered

color(red)(star) Sine Function : f(x) = sin (x)
color(red)(star) Cosine Function : f(x) = cos (x)
color(red)(star) Tangent Function : f(x) = tan (x)
color(red)(star) Cotangent Function : f(x) = cot (x)
color(red)(star) Secant Function : f(x) = sec (x)
color(red)(star) Cosecant Function : f(x) = cosec (x)

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

color(red)(✍️ ul "Domain :") all real numbers

color(green)(✍️ ul "Range :") " "[-1 , 1]

color(red)(✍️ ul "Period = ") 2pi

color(green)(✍️ ul "x intercepts : ") color(blue)(x = k pi) , where k is an integer.

color(red)(✍️ ul "y intercepts : ") color(blue)(y = 0)

color(green)(✍️ ul "maximum points :") color(blue)((pi/2 + 2 k pi "," 1)) , where k is an integer.

color(red)(✍️ ul "Minimum points :") color(blue)(((3pi)/2 + 2 k pi "," -1)) , where k is an integer.

color(green)(✍️ ul "Symmetry:") since color(blue)(sin(-x) = - sin (x)) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0 , 0).

color(red)(✍️ ul "Iintervals 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)

color(red)(✍️ ul "Domain :") all real numbers

color(green)(✍️ ul "Range :")  [-1 , 1]

color(red)(✍️ ul "Period = ")  2pi

color(green)(✍️ ul "x intercepts : ") x = pi/2 + k pi , where k is an integer.

color(red)(✍️ ul "y intercepts : ") y = 1

color(green)(✍️ ul "maximum points :") (2 k pi , 1) , where k is an integer.

color(red)(✍️ ul "Minimum points :")  (pi + 2 k pi , -1) , where k is an integer.

color(green)(✍️ ul "Symmetry:") since cos(-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to the y axis.

color(red)(✍️ ul "Iintervals 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)

color(red)(✍️ ul "Domain :") all real numbers except pi/2 + k pi, k is an integer.

color(green)(✍️ ul "Range :") all real numbers

color(red)(✍️ ul "Period = ")  pi

color(green)(✍️ ul "x intercepts : ") x = k pi , where k is an integer.

color(red)(✍️ ul "y intercepts : ") y = 0

color(green)(✍️ ul "Symmetry:") since tan(-x) = - tan(x) then tan (x) is an odd function and its graph is symmetric with respect the origin.

color(red)(✍️ ul "Iintervals of increase/decrease: ") over one period and from -pi/2 to pi/2, tan (x) is increasing.

color(green)(✍️ ul " Vertical asymptotes : ")  x = pi/2 + k pi, where k is an integer.

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

color(red)(✍️ ul "Domain :") all real numbers except k pi, k is an integer.

color(green)(✍️ ul "Range :") all real numbers

color(red)(✍️ ul "Period = ")  pi

color(green)(✍️ ul "x intercepts : ")  x = pi /2 + k pi , where k is an integer.

color(green)(✍️ ul "Symmetry:") since cot(-x) = - cot(x) then cot (x) is an odd function and its graph is symmetric with respect the origin.

color(red)(✍️ ul "Iintervals of increase/decrease: ") over one period and from 0 to pi, cot (x) is decreasing.

color(green)(✍️ ul " Vertical asymptotes : ")  x = k pi , where k is an integer.

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

color(red)(✍️ ul "Domain :") all real numbers except pi/2 + k pi , n is an integer.

color(green)(✍️ ul "Range :")  (-oo , -1] U [1 , +oo)

color(red)(✍️ ul "Period = ") 2 pi

color(green)(✍️ ul "y intercepts : ")  y = 1

color(green)(✍️ ul "Symmetry:") since sec(-x) = sec (x) then sec (x) is an even function and its graph is symmetric with respect to the y axis.

color(red)(✍️ ul "Iintervals 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) .

color(green)(✍️ ul " Vertical asymptotes : ")  x = pi/2 + k pi, where k is an integer.

Cosecant Function : f(x) = cosec (x)

color(red)(✍️ ul "Domain :") all real numbers except k pi, k is an integer.

color(green)(✍️ ul "Range :")  (-oo , -1] U [1 , +oo)

color(red)(✍️ ul "Period = ")  2pi

color(green)(✍️ ul "Symmetry:") since csc(-x) = - csc(x) then csc (x) is an odd function and its graph is symmetric with respect the origin.

color(red)(✍️ ul "Iintervals 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).

color(green)(✍️ ul " Vertical asymptotes : ") x = k pi, where k is an integer.