Transcript Ask yourself... how can I use the special angles from my unit circle to

Section 10.4: Sum and Difference Formulas
If I asked you to find an exact value of
trigonometric function without a calculator,
which angles could you use?
The sum and difference formulas allow you to
add or subtract any two of the special angles
found on our unit circle WITHOUT a calculator.
sin (u ± v) = sin u cos v ± cos u sin v
(same sign)
cos (u ± v) = cos u cos v ∓ sin u sin v
(opposite sign)
Ex. 1
Find the exact value of cos 75
(Ask yourself... how can I use the special angles
from my unit circle to add or subtract and yield
75?)
How about 45 + 30 ?
cos 75 = cos 45 cos 30 - sin 45 sin 30
π.
Ex. 2. Find the exact value of the sin12
(Ask yourself...how can I use the special angles
from my unit circle to add or subtract and yield
π/12? I know this involves fractions, but they
are my friend.)
Tip: This problem is in 12ths. So, look at the
angles on your unit circle in terms of 12ths
versus their reduced form.
π π
How about 3 - 4 ?
π
sin =
12
Ex. 3: Let’s Try A Tangent Problem!
Find the exact value of
tan

12
tan u  tan v
tan(u  v) 
1 tan u tan v

  
tan  tan   
12
3 4
Ex. 4: Let’s go backward!
Find the exact value of:
sin 42 cos12  cos 42 sin12
10.4 Learning Opportunity
Read Section 10.4
p. 653 #1-35 odd
10.5 Double and Half Angle Formulas
In this section, we will continue to add to our
identities. So far we have learned:
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Cofunction Identities
Even/Odd Identities
Sum and Difference Identities
Double Angle Formulas
sin 2u  2sin u cos u
cos 2u  cos u  sin u
2
2
 2 cos 2 u  1
 1  2sin 2 u
2 tan u
tan 2u 
1  tan 2 u
Double Angle Formulas

1  cos 
sin  
2
2

1  cos 
cos  
2
2

The sign
depends on the
quadrant where
θ/2 is located
1  cos 
sin 
tan 

2
sin 
1  cos 
Ex. 1
Use the following to find sin 2θ, cos 2θ, tan 2θ
5 3
cos   ,
   2
13 2
Ex. 2
 
 
 
Use the following to find: sin   , cos   , tan  
2
2
2
5 3
cos   ,
   2
13 2
Ex. 3
Use the figure to find the exact value of the trigonometric
function.
a. tan θ
b. sin 2θ
c. sec 2θ
1
θ
4
d. cot 2θ
Ex. 4
Use the figure to find the exact value of the
trigonometric function.
sin

sec
2

2
8
θ
15

cot
2
10.5 Learning Opportunity
Please Read Section 10.5
p. 660-661
#1-7 odd,
23-27 odd,
35-39 odd,
49-53 odd