Transcript Sum and Difference Formulas

Sum and Difference
Formulas
New Identities
Cosine Formulas
cos      cos  cos   sin  sin 
cos      cos  cos   sin  sin 
Sine Formulas
sin      sin  cos   cos  sin 
sin      sin  cos   cos  sin 
Tangent Formulas
tan   tan 
tan     
1  tan  tan 
tan   tan 
tan     
1  tan  tan 
Using Sum Formulas to Find Exact
Values



Find the exact value of cos 75o
cos 75o = cos (30o + 45o)
cos 30o cos 45o – sin 30o sin 45o
3 2 1 2

 

2 2 2 2
6 2
1
or
6 2
4
4


Find the Exact Value

Find the exact value of
7
sin
12
Change to degrees first (easier to find angles)
7 180

 105
12 
sin(105 )  sin  60  45 
sin 60 cos 45  sin 45 cos 60
1 2
2 3
2
6 1





or
2 2
2 2
4
4
4

2 6

Exact Value

Find the exact value of tan 195o
tan 45  tan150
tan(45  150 ) 
1  tan 45 tan150


1
1
1
1


3
3
3 



1 
1 

3 
1  1  
1  1  



3
3



 
3 1

3 1
Using Difference Formula to Find
Exact Values

Find the exact value of
sin 80o cos 20o – sin 20o cos 80o

This is the sin difference identity so . . .

 sin(80o
–
20o)
= sin
(60o)
=
3
2
Using Difference Formula to Find
Exact Values

Find the exact value of

cos 70o cos 20o – sin 70o sin 20o

This is just the cos difference formula

cos (70o + 20o) = cos (90o) = 0
Finding Exact Values
4 
If it is known that sin = ,     , and that
5 2
2
3
sin  = 
,   
, find the exact value of
2
5
a. cos ( + )
b. sin ( + )
c. tan (   )
Establishing an Identity

Establish the identity:
cos(   )
 cot  cot   1
sin  sin 
cos  cos   sin  sin 
 cot  cot   1
sin  sin 
cos  cos  sin  sin 

 cot  cot   1
sin  sin  sin  sin 
cot  cot   1  cot  cot   1
Establishing an Identity

Establish the identity

cos (    cos ( –   2 cos  cos 
Solution




cos (    cos ( –   2 cos  cos 
cos  cos  – sin  sin  + cos  cos   sin  sin 
cos  cos   cos  cos   2 cos  cos 
2 cos  cos  = 2 cos  cos 
Establishing an Identity


Prove the identity:
tan (q   = tan q
Solution
tan q  tan 
tan(q   ) 
1  tan q tan 
tan q  0

1  tan q  tan 0
tan q

 tan q
1
Establishing an Identity

Prove the identity:


tan  q     cot q
2

Solution
Since tan

2
sinq
tan q =
cosq
is undefined we have to use the identity


sin  q   sin q cos   cos q sin 

2


2
2
tan  q   




2


cos  q   cos q cos  sin q sin
2
2
2

sin q  0  cos q 1 cos q


  cot q
cos q  0  sin q 1  sin q
Finding Exact Values Involving
Inverse Trig Functions

Find the exact value of:
Solution


Think of this equation as the cos (  
Remember that the answer to an inverse
trig question is an angle).
So . . .  is in the 1st quadrant and  is in
the 4th quadrant (remember range)
Solution
cos(   )  cos  cos   sin  sin 
 5
tan    
 12 
 5 y
tan     
 12  x
y  5; x  12 need r
 3
sin     
 5
 3 y
sin      
 5 r
y  3;
r  5 need x
r  12  5
5  x   3
1
2
2
1
2
2
r  144  25  13
25  x  9
12
5
cos   ; sin  
13
13
4
cos  
5
2
2
16  x
2
x4
Solution
 12  4   5  3 
cos            
 13  5   13  5 
 48   15 
   
 65   65 
33

65
Writing a Trig Expression as an
Algebraic Expression



Write sin (sin-1u + cos-1v) as an
algebraic expression containing u and
v (without any trigonometric functions)
Again, remember that this is just a
sum formula
sin (   = sin  cos  + sin  cos 
Solution


Let sin-1u =  and cos-1v = 
Then sin   u and cos  = v
cos   1  sin 2   1  u 2
sin   1  cos 2   1  v 2
So
sin(sin 1 u  cos 1 v)  sin    
 sin  cos   cos  sin 
 uv  1  u 2 1  v 2
On-line Help

Tutorials

Videos