Download Trigonometric Formulas

Trigonometric Formulas
Definitions of trigonometric functions on the unit circle ( x 2  y 2  1 ):
cos t  x
sin t  y
csc t 
1
, y0
y
sec t 
1
, x0
x
y
, x0
x
x
cot t  , y  0
y
tan t 
Definitions of trigonometric functions on the right triangle:
opposite
hypotenuse
hypotenuse
csc  
opposite
sin  
adjacent
hypotenuse
hypotenuse
sec  
adjacent
cos  
opposite
adjacent
adjacent
cot  
opposite
tan  
Reciprocal relations:
1
sin t
sin t
tan t 
cos t
csc t 
sec t 
1
cos t
1
tan t
cos t
cot t 
sin t
cot t 
Even-Odd properties:
sin  t    sin t [odd]
csc t    csc t [odd]
cos t   cos t [even]
sec t   sec t [even]
tan  t    tan t [odd]
cot t    cot t [odd]
Pythagorean identities:
sin 2 t  cos 2 t  1
tan 2 t  1  sec 2 t
1  cot 2 t  csc 2 t
Note:
sin 2 t  cos 2 t  1 
sin 2 t  cos 2 t
1

 tan 2 t  1  sec2 t
2
cos t
cos 2 t
and
sin 2 t  cos 2 t
1
sin t  cos t  1 

 1  cot 2 t  csc2 t
2
2
sin t
sin t
2
2
Trigonometric Identities
Addition and Subtraction formulas:
sin s  t   sin s cost   coss sin t 
coss  t   coss cost   sin s sin t 
tan s   tan t  sin s  t 
tan s  t  

1  tan s  tan t  coss  t 
sin s  t   sin s cost   coss sin t 
coss  t   coss cost   sin s sin t 
tan s   tan t  sin s  t 
tan s  t  

1  tan s  tan t  coss  t 
Double Angle formulas:
sin 2x  2 sin xcosx
cos2 x   cos 2 x   sin 2 x   1  2 sin 2 x   2 cos 2 x   1
2 tan x 
sin 2 x 
tan 2 x  

2
1  tan  x  cos2 x 
Half Angle formulas:
1  cos2 x 
2
2 x 
1

cos
cos 2 x  
2
sin 2 x  
The above eleven (11) Trigonometric Identities (as well as many other trigonometric identities) can be derived
from DeMoivre’s Formula:
DeMoivre’s Theorem:
p 
,  cos   i sin    cos( p )  i sin( p )
p
For p = 2:
 cos   i sin  
2
 cos(2 )  i sin(2 ) .
 cos2   2i cos  sin   sin 2   cos(2 )  i sin(2 )
 cos2   sin 2   i  2cos sin    cos(2 )  i sin(2 )
So, cos2   sin 2   cos(2 ) and 2 cos  sin   sin(2 )
For p = 3:
 cos   i sin  
3
 cos  3   i sin  3 
 cos3   3i cos2  sin   3cos sin 2   i sin3   cos 3   i sin 3 
 cos3   3cos  sin 2   i 3cos 2  sin   sin 3    cos(3 )  i sin(3 ) .
So, cos3   3cos sin 2   cos(3 ) and 3cos2  sin   sin 3   sin(3 ) .
You can further rearrange these identities using the Pythagorean identities.
For Double-Angle Formula for tan  2x  :
2sin  x  cos  x 
sin  2 x 
2sin  x  cos  x 
cos 2  x 
2 tan  x 
tan  2 x  



2
2
2
2
cos  2 x  cos  x   sin  x  cos  x   sin  x  1  tan 2  x 
cos 2  x 
For half-angle formulas:
Rewrite double-angle formulas. For example,
cos 2   sin 2   cos(2 )  cos 2   1  cos 2    cos(2 )
 2 cos 2   1  cos(2 )  cos 2  
1  cos(2 )
2
For Addition Formulas:
 cos  s   i sin  s  cos t   i sin t   cos(s  t )  i sin(s  t )
 cos  s  cos  t   i cos  s  sin t   i sin  s  cos t   sin  s  sin t   cos(s  t )  i sin(s  t )
 cos  s  cos  t   sin  s  sin  t   i cos  s  sin  t   sin  s  cos  t    cos( s  t )  i sin( s  t ) .
So, cos  s  cos  t   sin  s  sin t   cos(s  t ) and cos  s  sin  t   sin  s  cos  t   sin(s  t ) .
For tan  s  t  :
cos  s  sin  t   sin  s  cos  t 
sin  s  t  cos  s  sin  t   sin  s  cos  t 
cos  s  cos  t 
tan  t   tan  s 
tan  s  t  



cos  s  t  cos  s  cos  t   sin  s  sin  t  cos  s  cos  t   sin  s  sin  t  1  tan  s  tan  t 
cos  s  cos  t 
For Subtraction Formulas:
Use the Odd and Even Properties. For example,
cos  s  t   cos  s  (t )   cos  s  cos  t   sin  s  sin  t   cos  s  cos t   sin  s  sin t  .
Trigonometric Functions
Function
f ( x)  sin( x)
Domain
 ,  
Range
[ 1,1]
f ( x)  cos( x)
 ,  
[ 1,1]
f ( x)  tan( x)



 x | x   n , n  
2


 ,  
f ( x)  cot( x)
x | x  n , n  
x | x  n , n  
 ,  
 , 1] [1,  
f ( x)  sec( x)



 x | x   n , n  
2


 , 1] [1,  
f ( x)  sin 1 ( x)
 arcsin( x)
[ 1,1]
  
  2 , 2 
f ( x)  cos 1 ( x)
 arccos( x)
[ 1,1]
[0,  ]
f ( x)  tan 1 ( x)
 arctan( x)
 ,  
  
 , 
 2 2
f ( x)  csc( x)
Graph
Trigonometric Ratios of the Reference Angles
Recognize the patterns in each column.
Reference
Angles
sin(t)
cos(t)
0°
0
0
2
4
1
2

6
30°
1 1

2
2
3
2

4
45°
2
2
2
2

3
60°
3
2
1 1

2
2

2
90°
4
1
2
0
0
2
t
0
tan(t)
cot(t)
csc(t)
sec(t)
θ
0
2  0 0
1
4
2
undefined
1
2  1  3
3
3
3
2
2
2 1
2
2
3
2  3 3
1
1
2
4
2 1
0
0
2
 undefined
3
1
0

1

2
2
1

2
2
3
3

2
1
2
1
3
4
2 3
3
2
2
1
0
1
2
 undefined
1
2
1
2
0

2
3
2

2
1
2
2 3
3
2
undefined
Evaluating Trigonometric Functions
Steps:
1. Identify the quadrant for given angle:
i. (0, 0.5π) is Quadrant I
ii. (0.5π, π) is Q II
iii. (π, 1.5π) is Q III
iv. (1.5π, 2π) is Q IV
2. Use definitions of trigonometric functions on the unit circle, to determine if answer is + or –.
3. Identify the reference angle using first column of the above table.
4. Identify the ratio using the appropriate column and row.
 7 
Example: Evaluate cos 

 4 
1. 7/4 = 1.75 which is between 1.5 and 2, so the angle is in Q IV;
2. cos t  x ; x > 0 in Q IV;
3. Reference angle is π/4 (using the denominator of given angle);
2
 7 
 
4. So, cos 
.
   cos    
2
 4 
4
Evaluating Trigonometric Functions
Steps:
1. Use the given ratio and the appropriate column and row of the above table to identify reference angle:
2. Use sign and the domain and range of inverse trigonometric function to determine the quadrant.
3. Use reference angle and quadrant to find the angle:
i. Quadrant I: angle = reference angle
ii. Q II: angle = π – reference angle
iii. Q III: angle = π + reference angle
iv. Q IV: angle = 2π – reference angle

Example: Evaluate tan 1  3

1. 3 in the tan(t) column is in the π/3 row;
2. tan t  0 in Q IV;


3. Angle is 2π – π/3 = 5π/3. Thus, tan 1  3 
5
.
3
Solving Trigonometric Equations
Steps:
1. Move all terms to one side of equation;
2. Factor and set each factor to zero [i.e., Use the Zero-Product Property of Real Numbers];
3. Rearrange each equation in Step 2 and solve by evaluating inverse trigonometric functions;
4. Use period of the trigonometric function to find all solutions.
Example: Solve sin  2 x   cos( x)
1. sin  2 x   cos( x)  0 ;
2. sin  2 x   cos( x)  0  2sin( x)cos( x)  cos( x)  0  cos( x) 2sin( x) 1  0 ;
3. cos( x)  0  x 
 3
,
2 2
3
 n(2 ), n 
4. x   n(2 ), n  ;
2
2

Example: Solve sin  2t  
1
 5
x ,
.
2
6 6

5
 n(2 ), n 
and x   n(2 ), n  ;
6
6
and 2sin( x)  1  0  sin( x) 
2
2
Simplifying Trigonometric Expressions
cos2 ( x)sin( x)  sin 3 ( x)
Example: Simplify
.
cos( x)
Example: Simplify
sin(2 x)
.
cos( x)
Practice
Evaluate.
 3 
1. sin  
 4 
 5 
2. sec 

 3 
5. csc1  2 
6. cot 1  1
3. cos  
 7 
4. tan 

 5 
3. sin  2   sin( )
4. cos  3   1
Find ALL Real Solutions to Trigonometric Equations
1. tan  t    3
2. sin  2t  
2
2