Download Trigonometric Graphs

1
Right Triangle:
sin 
opp
hyp
Right Triangle:
cos 
adj
hyp
Right Triangle:
tan 
opp
adj
Circular function, radius r
sin 
y
r
Circular function, radius r
cos 
x
r
2
Circular function, radius r
tan 
y
x
Reciprocal Identity:
csc 
1
sin 
Reciprocal Identity:
sec 
1
cos 
Reciprocal Identity:
cot 
Reciprocal Identity:
1
1
tan 
sin x csc x
or
cos x sec x
or
tan x cot x
3
Pythagorean Identities:
sin 2 x + cos 2 x =
1
Pythagorean Identities:
2
1 + tan x =
sec 2 x
Pythagorean Identities:
2
1 + cot x =
csc 2 x
Cofunction Identities
sin (/2 - x)
cos x
Cofunction Identities
cos (/2 - x)
sin x
4
Cofunction Identities
tan (/2 - x)
cot x
Cofunctions of
are equal.
Complementary Angles
Sum/Difference Formulas:
sin (u + v) =
sin u cosv  sin v cosu
Sum/Difference Formulas:
cos (u + v) =
Sum/Difference Formulas:
tan (u + v) =
cosu cosv
sin u sin v
tan u  tan v
1 tan u tan v
5
Double angle Formula
sin 2u
2 sin u cos u
Double angle Formula
cos 2 u - sin 2 u
2 cos 2 u - 1
1 - 2 sin 2 u
cos 2u
Double angle Formula
tan 2u
2 tan u
2
1  tan u
Power Reducing Formulas
2
sin u
1  cos 2u
2
Power Reducing Formulas
2
cos u
1  cos 2u
2
6
Domain
(-∞,∞)
Range
[-1,1]
Period 2π
Graph of y = sin x
Domain
(-∞,∞)
Range
[-1,1]
Period 2π
Graph of y = cos x
Domain
x
Graph of y = tan x

n
, n I
2
Range
(-∞,∞)
Period π
Domain
x
Graph of y = cot x

n
, n I
2
Range
(-∞,∞)
Period π
Domain
x
Graph of y = sec x

n
, n I
2
Range
|y| > 1
Period π
7
Power Reducing Formulas
2
tan u
1  cos 2u
1  cos 2u
Product-to-Sum:
sin u sin v
1
[cos(u  v)  cos(u  v)]
2
Product-to-Sum:
cos u cos v
Product-to-Sum:
sin u cos v
Product-to-Sum:
cos u sin v
1
[cos(u  v)  cos(u  v)]
2
1
[sin(u  v)  sin(u  v)]
2
1
[sin(u  v)  sin(u  v)]
2
8
Domain
x  n , n  I
Graph of y = csc x

Range
|y| > 1
Period 2π
Domain
[-1,1]
Range
Graph of y = sin-1 x
   
 ' 
 2 2 

Graph of y = cos-1 x
Domain
[-1,1]
Range
[0,π]
Domain
(-∞,∞)
Range
Graph of y = tan-1 x
   
 , 
 2 2 

Graph of y = sec-1 x
Domain
|x| >1
Range
[0,], y 


2
9
Domain
|x| > 1
Range
Graph of y = csc-1 x
   
 , , y  0
 2 2 

Graph of y = cot-1 x
Domain
(-∞,∞)
Range
(0,π)
Graph of y = ex
Domain
(-∞,∞)
Range
(0,∞)
Graph of y = ln x
Domain
(0,∞)
Range
(-∞,∞)
Quadratic Formula
b  b 2  4ac
x
2a
10
Law of Sines
Sin A Sin B Sin C


a
b
c
Law of Cosines
c  a  b  2abcosC
Pythagorean
Theorem
1
sec x
1
csc x
2
2
2
2
2
c a b
1
cos
x
1
sin
1
1
x
2
11
x
a a
y
x
a
y
a
a 
x y
a
x
log a x  log a y
a
x y
a
x y
a
xy
1
x
a
log a xy
12
1
1
1
1
sin x  cos x
tan x  cot x
1
1
csc x  sec x
log a x  log a y
log a x
b

2

2

2
x 
log a 
 

y 
b log a x
13
Change of Base Formula
log a x
Transformation of
Trigonometric Graphs
ln x
ln a
Vertical stretch or shrink:
reflection about x-axis
Vertical Shift
y  af (b(x  c))  d
y  af (b(x  c))  d
Horizontal stretch or shrink:
Horizontal Shift
reflection about y-axis
Point-Slope Equation
y  m(x  x1)  y1
Slope
General Linear Equation
rise y y2  y1
m


run x x2  x1
Ax  By  C
14
2
30-60-90
Right Triangle
60
30
3
45-45-90
Right Triangle
45
2

45

1
1
1