Download TRIGONOMETRY

Definitions of Trig Functions of Any Angle
TRIGONOMETRY
Let  be any angle in
the standard position
and let P ( x, y ) be a
point on the terminal
side of  . If r is the
distance from  0, 0 
Right Triangle Definition of Trig Functions
to  x, y  and
r  x 2  y 2  0, x  0, y  0
y
,
r
r
csc   ,
y
x
cos   ,
r
r
sec   ,
x
sin  
opposite
a

hypotenuse c
adjacent
b
cos  

hypotenuse c
opposite a
tan  

adjacent b
sin  
hypotenuse c

opposite
a
hypotenuse c
sec  

adjacent
b
adjacent b
cot  

opposite a
csc  
y
x
x
cot  
y
tan  
Reference Angles
Let  be an angle in standard position. Its reference
angle is the acute angle   formed by the terminal
side of  and the horizontal axis.
Degrees to Radians Formulas
If x is an angle in degrees and t is an angle in
Example:
  315
radians then:
   360  315  45

t
x
180t

 t
and x 
180 x
180

Sines, Cosines, and Tangents of Special Angles
The Signs of Trig Functions
sin 
cos 
tan 
QI
+
+
+
Q II
+


Q III


+
Q IV

+

Formulas and Identities
sin 
cos 
cot  
Ratio: tan  
cos 
sin 
Reciprocal Identities
Even/Odd

0
30 
sin 
0
cos 
1
tan 
3
2
0
3
3
1
2

6
45 
2
2
2
2
1

4
60 
3
2
1
2
3

90 
3
1
0

2
1
sin 
1
sec  
cos 
1
cot  
tan 
csc  
cos( )  cos 
sin( )   sin  
 Odd
tan( )   tan  
dne
Pythagorean Identities
sin 2   cos 2   1
tan 2   1  sec 2 
1  cot 2   csc 2 
Even
TRIGONOMETRY
Formulas and Identities
Double Angle Formulas
sin(2 )  2sin  cos 
Graphs of Trig Functions
cos(2 )  cos 2   sin 2 
y  sin x
Domain: x  ,  
Range: y  [ 1,1]
Period: 2
Amplitude: 1
y  cos x
Domain: x  ,  
Range: y  [ 1,1]
Period: 2
Amplitude: 1
y  tan x
Domain:
x
and x 

 n
2
Range: y   ,  
Period: 
Amplitude: None
y  cot x
Domain: x  and x  n
Range: y   ,  
Period: 
Amplitude: None
y  sec x
Domain:
x
and x 
 1  2sin 2 
tan(2 ) 
2 tan 
1  tan 2 
Cofunction Formulas
sin(90   )  cos 
sin
2
2
1  cos 
2
2
 1  cos 
sin 
tan 

2
sin 
1  cos 
cos



cos(90   )  sin 




sin      cos 
cos      sin 
2

2

Sum and Difference Formulas
sin(   )  sin  cos   cos  sin 
cos(   )  cos  cos  sin  sin 
tan   tan 
tan(   ) 
1 tan  tan 
Sum to Product Formulas
   
   
sin   sin   2sin 
 cos 

 2 
 2 
       
sin   sin   2 cos 
 sin 

 2   2 
   
   
cos   cos   2 cos 
 cos 

 2 
 2 
       
cos   cos   2sin 
 sin 

 2   2 
Product to Sum Formulas

2
 n
Range:
y  (, 1]  [1, )
Period: 2
Amplitude: None
y  csc x
Domain: x 
Range:
 2 cos 2   1
Half-Angle Formulas

1  cos 
and x  n
y  (, 1]  [1, )
Period: 2
Amplitude: None
1
cos(   )  cos(   ) 
2
1
cos  cos    cos(   )  cos(   ) 
2
1
sin  cos   sin(   )  sin(   ) 
2
1
cos  sin   sin(   )  sin(   ) 
2
sin  sin  
Inverse Trig Functions
y  sin 1 x is equivalent to x  sin y,


 y

y  cos x is equivalent to x  cos y,
2
2
0 y 
y  tan 1 x is equivalent to x  tan y,

1

2
 y

2