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.

**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).

**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).

**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.

**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.

**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.

**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