4.6 Graphs of Other Trigonometric Functions

Graph of Tangent Functions

• Remember tangent is an odd function
• tan(-x) = -tan(x)
• This means that the graph will be symmetric with respect to the origin
• tan x = sin x/cos x
• tan x is undefined when cos x = 0
• cos x = 0 when x=π/2 and x=-π/2
• Thus y=tan x has vertical asymptotes at π/2 and -π/2
• Since we know the period is π there are also asymptotes at x=π/2 + πn where n is an integer
• Period of tangent and cotangent is π/b
• In order to sketch the graph locate key points, identify intercepts and asymptotes, it is usually best to sketch more than one period.

• Period = π
• Domain =  all
• Range = 
• Vertical Asymptotes = 

Graph of Cotangent Functions

• y= cot x = cos x / sin x 
• Cotangent functions are similar to tangent functions however since sin x = 0 makes the cot x undefined the asymptotes are at x = nπ where n is an integer.
• Notice the graphs of tangent are increasing while the graphs of cotangent are decreasing

 

• Period = π
• Domain = all 
• Range =
• Vertical Asymptotes = 

Graphs of Reciprocal Functions

• 1/sin x = csc x
• 1/cos x = sec x 
• These graphs are found by using the reciprocal identities. These graphs also have asymptotes because for sec x for example it is undefined when cos x = 0. 
• Also for given x, the y coordinates of sec x are the reciprocal of the y coordinates of cos x.
• In order to graph the reciprocals csc x and sec x you must first graph the reciprocal functions which are sin x and cos x receptively, keep in mind the asymptotes.
• You should get a u-shaped result, however it is not a parabola.

Cosecant Graph

• Period = 2π
• Domain = all
• Range = 
• Vertical Asymptotes =  
• Symmetry= y-axis 

Secant Graph

 

• Period = 2π
• Domain = all 
• Range = 
• Vertical Asymptotes = 
• Symmetry = origin

If you compare sine and cosine functions to their respective reciprocals you can notice in the graphs that the hills and valleys are interchangeable. For example the hill of the max point on a sine curve is the valley, or low point on a cosecant curve and vice versa.

Shifting Graphs

• To shift the graphs of the parent functions
• Use the same formula
• y = a*tan(b(x-c))+d
• Keep in mind the period for tangent and cotangent is π/b 
• Also amplitude for tangent and cotangent is not necessarily important if the y axis isn't labeled
• Also c is a phase shift and thus also effects asymptotes.
• See section 4.5 for shifts of sin and cosine and definitions of each of the letters

All the graphs in one