Properties of the Six Trigonometric Functions
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
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