Section 4.6
Graphs of Other Trigonometric
Functions
Overview
• In this section we examine the graphs of the
other four trigonometric functions.
• After looking at the basic, untransformed
graphs we will examine transformations of
tangent, cotangent, secant, and cosecant.
• Again, extensive practice at drawing these
graphs using graph paper is strongly
recommended.
Tangent and Cotangent
• Three key elements of tangent and cotangent:
1. For which angles are tangent and cotangent
equal to 0? These will be x-intercepts for your
graph.
2. For which angles are tangent and cotangent
undefined? These will be locations for vertical
asymptotes.
3. For which angles are tangent and cotangent
equal to 1 or -1? These will help to determine
the behavior of the graph between the
asymptotes.
y = tan x
y = cot x
Transformations
y  A tan( Bx  C )
y  A cot( Bx  C )
|A| = amplitude
π/B = period (distance between
asymptotes).
C/B gives phase shift from zero.
Examples—Graph the Following
x
y  3 tan
2


y  cot  x  
2

Secant and Cosecant
• The graphs of secant and cosecant are derived
from the graphs of cosine and sine, respectively:
1. Where sine and cosine are 0, cosecant and
secant are undefined (location of vertical
asymptotes).
2. Where sine and cosine are 1, cosecant and
secant are also 1.
3. Where sine and cosine are -1, cosecant and
secant are also -1.
y = csc x
(derived from graph of y = sin x)
y = sec x
(derived from graph of y = cos x)
Transformations
• To graph a transformation of cosecant or secant,
graph the transformation of sine or cosine,
respectively, then use the reciprocal strategy
previously discussed: |A| = amplitude (affects the places
y  A sec( Bx  C )
y  A csc( Bx  C )
where secant or cosecant is equal
to 1 or -1)
2π/B = period (distance between
asymptotes)
C/B = phase (horizontal) shift, left if
(+), right if (-)
Examples—Graph the Following
x 
3
y  csc  
2
4