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P. 404: 1-6 S
P. 404: 23-30 S
P. 404: 31-36 S
P. 404: 37-50 S
P. 405: 51-54 S
P. 405: 55-68 S
P. 405: 69-72 S
HW Supplement
1. In a single column, number your paper from
1 to 24. (Pretend this is a single column.)
1
13
2
14
3
15
4
16
5
17
6
18
7
19
8
20
9
21
10
22
11
23
12
24
2. Now tag a π to the end of each number.
1 
13 
2 
14 
3 
15 
4 
16 
5 
17 
6 
18 
7 
19 
8 
20 
9 
21 
10 
22 
11 
23 
12 
24 
3. Divide everything by 12.
1  /12
13  /12
2  /12
14  /12
3  /12
15  /12
4  /12
16  /12
5  /12
17  /12
6  /12
18  /12
7  /12
19  /12
8  /12
20  /12
9  /12
21  /12
10  /12
22  /12
11  /12
23  /12
12  /12
24  /12
4. Finally simplify all your fractions and figure
out what the point of all that was.
1  /12
13  /12
2  /12   / 6
14  /12  7 / 6
3  /12   / 4
15  /12  5 / 4
4  /12   / 3
16  /12  4 / 3
5  /12
17  /12
6  /12   / 2
18  /12  3 / 2
7  /12
19  /12
8  /12  2 / 3
20  /12  5 / 3
9  /12  3 / 4
21  /12  7 / 4
10  /12  5 / 6
22  /12  11 / 6
11  /12
23  /12
12  /12  
24  /12  2
You were very clever to
figure out that those
radian angle measures
belong on a new and
improved, yet very
crowded, unit circle. Fill in
the angle measures in
degrees (inner circle) and
in radians (middle circle).
Maybe you can see where
this lesson is going.
Write 15° as the sum or difference of two angles
from the original unit circle.
Write π/12 as the sum or difference of two
angles from the original unit circle.
Write 75° as the sum or difference of two angles
from the original unit circle.
Write 5π/12 as the sum or difference of two
angles from the original unit circle.
Objective:
1. To use sum and
difference formulas
to find the exact
value of trig
functions, simplify
expressions, and
solve equations
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Assignment:
P. 404: 1-6 S
P. 404: 23-30 S
P. 404: 31-36 S
P. 404: 37-50 S
P. 405: 51-54 S
P. 405: 55-68 S
P. 405: 69-72 S
HW Supplement
sine
cosine
tangent
sin(u  v)  sin u cos v  cos u sin v
sin(u  v)  sin u cos v  cos u sin v
cos(u  v)  cos u cos v  sin u sin v
cos(u  v)  cos u cos v  sin u sin v
tan u  tan v
tan(u  v) 
1  tan u tan v
tan u  tan v
tan(u  v) 
1  tan u tan v
Use a sum or difference formula to find the
exact value of sin(15°).
Use a sum or difference formula to find the
exact value of cos(π/12).
Use a sum or difference formula to find the
exact value of sin(5π/12).
Use a sum or difference formula to find the
exact value of cos(75°).
Use a sum or difference formula to find the
exact value of tan(15°).
Use a sum or difference formula to find the
exact value of tan(5π/12).
Now use the exact values
for sine and cosine to
find the coordinates of
each point (outer circle)
on the new, very
crowded unit circle.
For the sides of a right triangle:
Long Side
6 2
4
Short Side
6 2
4
Use a sum or difference formula to prove:


cos   x   sin x
2

Find the exact value of
cos 25 cos 20  sin 25 sin 20
Write the following as an algebraic expression.
sin  arctan1  arccos x 
Solve for x.

3


sin  x    sin  x 
2
2



 1

Let’s prove cos (u – v) = cos u cos v + sin u sin v
First we’ll set up a diagram:
• On a unit circle, construct
angles u and v such that
u>v
• Now construct the angle
u–v
• Assign variables for each
point
Let’s prove cos (u – v) = cos u cos v + sin u sin v
Now draw in two segments,
AC and BD.
• Notice these segments
create 2 isosceles
triangles: ΔOAC and ΔOBD
• Since <AOC is congruent to
<DOB, the two triangles
are congruent
• Thus, AC = BD
(Click for dynamic diagram.)
Let’s prove cos (u – v) = cos u cos v + sin u sin v
AC  BD
(By the distance formula)
 x2 1   y2  0
2
2

 x3  x1    y3  y1 
2
(By squaring both sides)
 x2  1   y2  0 
2
2
  x3  x1    y3  y1 
2
2
2
Let’s prove cos (u – v) = cos u cos v + sin u sin v
AC  BD
(By the distance formula)
 x2 1   y2  0
2
2

 x3  x1    y3  y1 
2
2
(By squaring both sides)
 x2  1   y2  0 
2
2
  x3  x1    y3  y1 
2
2
(By simplifying both sides)
x2 2  2 x2  1  y2 2  x32  2 x3 x1  x12  y32  2 y3 y1  y12
Let’s prove cos (u – v) = cos u cos v + sin u sin v
(Since point B, C, and D lie on the unit circle)
x2 2  y2 2  1
x32  y32  1
x12  y12  1
x2 2  2 x2  1  y2 2  x32  2 x3 x1  x12  y32  2 y3 y1  y12
(By Substitution)
2 x2  1  1  2 x3 x1  1  2 y3 y1  1
(By simplifying)
2 x2  2 x3 x1  2 y3 y1
Let’s prove cos (u – v) = cos u cos v + sin u sin v
2 x2  2 x3 x1  2 y3 y1
(By dividing both sides by 2)
(By unit circle definitions)
x2  x3 x1  y3 y1
cos  u  v   x2
cos u  x3
cos v  x1
sin u  y3
sin v  y1
(By Substitution)
cos  u  v   cos u cos v  sin u sin v
After proving cos (u – v) = cos u cos v + sin u sin v :
Prove cos (u + v):
1. Let u + v =
u – (−v)
2. Use substitution
and even/odd
identities:
Prove sin (u ± v):
1. Let u = π/2 – x
in cos (u ± v)
2. Use cofunction
identities to turn
cosine into sine:
cos   x   cos x
cos  2  x   sin x
sin   x    sin x
sin  2  x   cos x
Prove tan (u ± v):
1. Use quotient
identity:
tan x 
2.
sin x
cos x
Use substitution
with the sum
and difference
formulas for
sine and cosine
Objective:
1. To use sum and
difference formulas
to find the exact
value of trig
functions, simplify
expressions, and
solve equations
•
•
•
•
•
•
•
•
Assignment:
P. 404: 1-6 S
P. 404: 23-30 S
P. 404: 31-36 S
P. 404: 37-50 S
P. 405: 51-54 S
P. 405: 55-68 S
P. 405: 69-72 S
HW Supplement