The document summarizes the key properties of six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. It provides the period, x-intercepts, y-intercepts, and locations of any vertical asymptotes for each function.

1. TARUNGEHLOT
Graphs of Trigonometric Functions
Sine Cosine
Period = 2  Period = 2 
y = a sin (bx + c) y = a cos (bx + c)
amplitude = a amplitude = a
2 2
period = period =
b b
c c
phase shift = phase shift =
b b
one cycle can be found by solving: one cycle can be found by solving:
0  bx  c  2 0  bx  c  2
Tangent Cotangent
Period =  Period = 
x –intercepts at  n 
x –intercepts at  n
 2
vertical asymptotes at x =  n
2 vertical asymptotes at x =  n
y = a tan (bx + c) y = a cot (bx + c)

2.  
period = period =
b b
c c
phase shift = phase shift =
b b
successive vert. asymptotes for one branch: successive vert. asymptotes for one branch:
  2  bx  c   2 0  bx  c  
Cosecant Secant
Period = 2  Period = 2 
Vertical asymptotes at x =  n 
Vertical asymptotes at x =  n
2
y = a csc (bx + c) y = a sec (bx + c)
2 2
period = period =
b b
c c
phase shift = phase shift =
b b
one cycle can be found by solving: one cycle can be found by solving:
0  bx  c  2  3
 bx  c 
2 2
To graph y = a csc (bx + c):
First graph y = a sin (bx + c); draw the To graph y = a sec (bx + c):
vertical asymptotes at the x-intercepts; vertical asymptotes at the x-ints,
First graph y = a cos (bx + c); draw the
take the reciprocals. vertical asymptotes at the x-intercepts; vertical as
take the reciprocals.
Summary:
period x-intercepts y-intercepts Vertical asymptotes
y = sin x 2 n 0 none
y = cos x 2  1 none
 n
2
y = tan x  n 0 
x   n
2

3. y = cot x   none x  n
 n
2
y = sec x 2 none 1 
x   n
2
y = csc x 2 none none x  n