Identify the parent function
Secant: (y=sec x=frac{1}{cos x})
Cosecant: (y=csc x=frac{1}{sin x})
Find the period
For (y=asec(bx-c)+d) or (y=acsc(bx-c)+d), period (=frac{2pi}{|b|})
Find the phase shift
Phase shift (=frac{c}{b})
Find the vertical shift
Vertical shift (=d)
Find the vertical stretch or reflection
If (a<0), reflect over the (x)-axis
If (|a|>1), stretch vertically
If (0<|a|<1), compress vertically
Find the asymptotes
Secant asymptotes occur where (cos(bx-c)=0)
Cosecant asymptotes occur where (sin(bx-c)=0)
Solve for secant asymptotes
Set (cos(bx-c)=0)
Use (bx-c=frac{pi}{2}+kpi)
Solve for (x)
Solve for cosecant asymptotes
Set (sin(bx-c)=0)
Use (bx-c=kpi)
Solve for (x)
Find key points from the reciprocal graph
Secant uses cosine key points
Cosecant uses sine key points
For secant
Start with cosine points
Plot where cosine equals (1) or (-1)
Convert those points to secant points:
(cos x=1 Rightarrow sec x=1)
(cos x=-1 Rightarrow sec x=-1)
For cosecant
Start with sine points
Plot where sine equals (1) or (-1)
Convert those points to cosecant points:
(sin x=1 Rightarrow csc x=1)
(sin x=-1 Rightarrow csc x=-1)
Sketch the asymptotes first
Plot the key points next
Draw the branches
Secant branches open upward above (y=1) or downward below (y=-1)
Cosecant branches open upward above (y=1) or downward below (y=-1)
Use the range
Secant: (yle -1) or (yge 1)
Cosecant: (yle -1) or (yge 1)
With transformations:
(yle d-|a|) or (yge d+|a|)
Use the domain restrictions
Secant: exclude values where (cos(bx-c)=0)
Cosecant: exclude values where (sin(bx-c)=0)
To solve equations involving secant
Rewrite (sec x) as (frac{1}{cos x})
Isolate the trigonometric expression
Convert to cosine
Solve for (x)
To solve equations involving cosecant
Rewrite (csc x) as (frac{1}{sin x})
Isolate the trigonometric expression
Convert to sine
Solve for (x)
Check for extraneous solutions
Reject any solution that makes the denominator zero
Example form for secant graphing
(y=2secleft(x-frac{pi}{3}right)-1)
Example form for cosecant graphing
(y=-3csc(2x)+4)
Graphing steps summary
Find period
Find phase shift
Find vertical shift
Find asymptotes
Plot key points
Draw branches
Check range and domain
Solving steps summary
Rewrite as sine or cosine
Isolate the trig function
Solve the trig equation
Write the general solution
Check for extraneous solutions