6 Trigonometric Functions

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6 Trigonometric Functions

Cotangent

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).