Sum and Difference Formulas

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Sum and Difference Formulas
Section 5.4
Exploration:

Are the following functions equal?
a) Y = Cos (x + 2)
b) Y = Cos x + Cos 2
How can we determine if they are equal by looking at their
graphs?
Graph them using your calculator.
Exploration
a) Y = Cos (x + 2)
b) Y = Cos x + Cos 2
Y = Cos x + Cos 2
Y = (x
Cos
(x + 2)
Y = Cos
+ 2)
Y = Cos x + Cos 2
Sum and Difference Formulas

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
Sum and Difference Formulas

Tan (u + v) =
Tan u + Tan v
1 - Tan u Tan v

Tan (u – v) =
Tan u - Tan v
1 + Tan u Tan v
Sum and Difference Formulas

Before we continue, think about all of the angles you can
find a trig function without using a calculator:
Choose any 2 of these and a trig function:
Sum and Difference Formulas
Sum and Difference Formulas

To find the trig function of an angle using the formulas:
1)
Find 2 angles whose sum or difference is equal to the
angle you are trying to evaluate
2)
Put the two angles into the appropriate formula
3)
Evaluate the trig functions of the angles you know
4)
Simplify
Sum and Difference Formulas

Evaluate: Sin 15º
What two angles have a sum or difference of 15º?
→ 45º - 30º
Put these two angles in the appropriate formula:
→ Sin (45º - 30º) = Sin 45º Cos 30º - Cos 45º Sin 30º
Sum and Difference Formulas
Sin 45º Cos 30º - Cos 45º Sin 30º
Evaluate the trig functions
2
3

2
2

2 1
2
2
Simplify
6

4
2
6 2

4
4
Sum and Difference Formulas
7
Cos
12
7  
 
12 3 4
7
 
Cos
 Cos (  )
12
3 4
7




Cos
 Cos Cos - Sin Sin
12
3
4
3
4
Sum and Difference Formulas
7




Cos
 Cos Cos - Sin Sin
12
3
4
3
4
1
2
3
2
 


2 2
2
2
2 6

4
Sum and Difference Formulas

a)
b)
Evaluate the following functions.
6 2
Cos 75 
4
o

6 2
Sin

12
4
Sum and Difference Formulas
o
Cos 75 = Cos 45º Cos 30º - Sin 45º Sin 30º
2
3  2 1


2
2
2
2
6 2

4
Sum and Difference Formulas
Sin

12
 Sin

3
Cos
2
3


2
2
6 2

4

4
- Cos
1
 
2

3
Sin
2
2

4
Sum and Difference Formulas
Section 5.4
Sum and Difference Formulas

Evaluate the following functions.
2

6
a) Cos - 105 
4
o
5

b) Tan
12
 32
Sum and Difference Formulas
Cos - 105 = Cos 150º Cos 45º + Sin 150º Sin 45º
o
3
1
2
2

 
2
2
2
2
2 6

4
Sum and Difference Formulas
5
 
Tan
 Tan (  )
12
6 4


T an ( )  T an( )
6
4


1 - T an ( )T an( )
6
4
3
3 3
1
3
3


3
3 3
13
3
Sum and Difference Formulas
3 3
3 3
5

Tan

12
3 3
3- 3
12 6 3

6
 2 3
Sum and Difference Formulas

Yesterday:



Used the formulas to evaluate trig functions of different angles
Worked with both radians and degrees
Today



Use the formulas to simplify longer expressions
Use the formulas to evaluate expressions from triangles
Use the formulas to create algebraic expressions
Sum and Difference Formulas

Find the exact value of the following expression:
Cos 78ºCos18º + Sin 78ºSin18º = ½
What formula is being used here?
→ Cos (u – v)
Re-write the expression using the formula
→ Cos (78º – 18º) = Cos 60º = ½
Sum and Difference Formulas

Use the sum and difference formulas to evaluate the
following:
o
o
o
o
a) Sin 42 Cos12 - Cos 42 Sin 12
o
 Sin 30  1
2




b) Cos Cos - Sin Sin
7
5
7
5
 
12
 Cos (  )  Cos
7 5
35
Sum and Difference Formulas

Find the exact value of the Cos (u – v) using the given
information:
7
4
Sin u = 
Cos v =
Both u and v and in quadrant III
25
5
When you are given 2 different criteria, you must draw 2
different triangles
-7
-24
u
25
-3
-4
v
5
Sum and Difference Formulas
-7
-24
u
-4
v
-3
25
5
Cos (u – v) = Cos u Cos v + Sin u Sin v

 24
4
7


25
5
25
96

125

21
117

125
125

3
5
Sum and Difference Formulas

Find the exact value of the trig functions given the
following information:
3
Tan u = 
4
13
Csc v = 
5
and both u and v are in quadrant IV.
 56

Find a) Sin (u + v)
65
65
b) Sec (u – v)

63
63
c) Cot (u – v)  
56
Sum and Difference Formulas
u
4
5
-3
v
12
13
Sin (u + v) = Sin u Cos v + Cos u Sin v
3
12
4
5




5
13
13
5
 36  20  56



65
65
65
-5
Sum and Difference Formulas
u
4
5
Sec (u - v) =
-3
v
1 ÷ Cos (u - v) 
12
13
65
63
Cos (u - v) = Cos u Cos v + Sin u Sin v
4
12
3



13
5
5
48
15
63



65
65
65
5

13
-5
Sum and Difference Formulas
u
4
5
-3
v
12
13
Cot (u - v) = 1 ÷ Tan (u - v)  
-5
63
56
Tan u - Tan v
Tan (u - v) =
1  Tan u Tan v
3 5
 36  20
 56


48  48
 4 12
 48
3 5
48  15
63
1


4 12
48 48
48
 56

63
Sum and Difference Formulas

Lastly, we would like to apply the process used in drawing
triangles to create algebraic expressions.

Same steps as before, just using variables instead of
numbers.
Sum and Difference Formulas

Write Cos (arcTan 1 + arcCos x) as an algebraic
statement.
→ What formula is being used?
Cos (u + v)
u = arcTan 1
v = arcCos x
Tan u = 1
Cos v = x
→ Use this information to draw your triangles.
Sum and Difference Formulas
Tan u = 1
2
u
Cos v = x
1
1
v
1
Cos (u + v) = Cos u Cos v – Sin u Sin v
1
1

x 
 1  x2
2
2
x  1 x2
1 x2
x



2
2
2
1  x2
x
Sum and Difference Formulas

Write the trig expression as an algebraic expression:
Sin (arcTan 2x – arcCos x)
Sin (u – v)
u = arcTan 2x
v = arcCos x
Tan u = 2x
Cos v = x
Sum and Difference Formulas
Cos v = x
Tan u = 2x
1  4x2
u
1
2x
v
1
Sin (u – v) = Sin u Cos v – Cos u Sin v


2x
x 
1  4x2
2x2
1  4x
2
1

1  4x2
1 x2
1 4x
2

 1  x2
2x2  1 x2
1 4x2
1  x2
x
Sum and Difference Formulas
Section 5.4
Sum and Difference Formulas

Write the trig expression as an algebraic expression:
Cos (arcSin 3x + arcTan 2x)
Cos (u + v)
u = arcSin 3x
v = arcTan 2x
Sin u = 3x
Tan v = 2x
Sum and Difference Formulas
Tan v = 2x
Sin u = 3x
1
1  4x2
3x
u
v
1  9x
2
Cos (u + v) = Cos u Cos v – Sin u Sin v
 1  9x

1 9x2
1 4x

2
2

1
1 4x
6x2
1  4x
2
 3x 
2

2x
1 4x2
1 9x2  6x2
1  4x2
1
2x
Sum and Difference Formulas

a)
b)
c)
d)
So far, in this section we have:
Used sum and difference formulas to evaluate trig
functions of different angles
Recognized sum and difference formulas to simplify
expressions
Used criteria to draw triangles and apply formulas
Create algebraic expressions
Lastly, we are going to simplify, verify, and solve equations
Sum and Difference Formulas

Simplifying:
o
Apply the formula
o
Evaluate trig functions that you know
o
Reduce the expression
Sum and Difference Formulas

Simplify the following expressioni:
Sin (90º – x)
→ Sin 90º Cos x – Cos 90º Sin x
→ (1) (Cos x) - (0) (Sin x)
= Cos x
Sum and Difference Formulas

Simplify the following expressioni:
Cos (x + 3π)
→ Cos x Cos 3π – Sin x Sin 3π
→ (Cos x)(0) - (Sin x) (1)
= Sin x
Sum and Difference Formulas

Verifying

Same process and simplifying

You are given what the expression should simplify to

As before, only work with 1 side of the equal sign
Sum and Difference Formulas

Verify the following identities:
a)
Tan (π + x) = Tan x
b)
Sin (x + y) Sin (x – y) = Cos² y – Cos² x
Sum and Difference Formulas
Tan (π + x) = Tan x
Tan   Tan x

1  Tan  Tan x
Tan x

1
 Tan x
0  T an x

1  (0) T an x
Sum and Difference Formulas
Sin (x + y) Sin (x – y) = Cos² y – Cos² x
= (Sin x Cos y + Cos x Sin y) ( Sin x Cos y – Sin x Cos y)
= Sin² x Cos² y - Cos² x Sin² y
= (1 - Cos² x) Cos² y - Cos² x (1 – Cos² y)
= Cos² y - Cos² x Cos² y - Cos² x + Cos² y Cos² x
= Cos² y – Cos² x
Sum and Difference Formulas

The last step in this section is using the sum and
difference formulas to solve equations.

Again, apply the formula, simplify, and now solve.
Sum and Difference Formulas
Cos (x 
Cos x Cos
Cos x Cos

4

4


) - Cos (x - )  1
4
4
 Sin x Sin
 Sin x Sin

4
- (Cos x Cos

- Cos x Cos
4
 2Sin x Sin

4
1

4

4
 Sin x Sin
 Sin x Sin

4

4
)
)
Sum and Difference Formulas
Cos (x 


) - Cos (x - )  1
4
4

 2
 2Sin x Sin
Sin x  1
1
 2

4
2


1
 2Sin x  1
Sin x  2
5 7
x
,
4
4
Sum and Difference Formulas
Sin (x 
Sin x Cos

3

3
)  Sin (x 
 Cos x Sin

3
) 1
 Sin x Cos
3
2Sin x Cos


3
1

3
 Cos x Sin

3
Sum and Difference Formulas
Sin (x 
2Sin x Cos
Sin x  1

3

3
)  Sin (x 

3
) 1
1
2 Sin x  1
2
1

x
2

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