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.

Explorationa) Y = Cos (x + 2) b) Y = Cos x

+ Cos 2

Y = Cos (x + 2)Y = Cos x + Cos 2

Y = Cos x + Cos 2Y = Cos (x + 2)

Sum and Difference Formulas

Sin (u + v) =

Sin (u – v) =

Cos (u + v) =

Cos (u – v) =

Sin u Cos v + Cos u Sin v

Sin u Cos v – Cos u Sin v

Cos u Cos v – Sin u Sin v

Cos u Cos v + Sin u Sin v

Sum and Difference Formulas

Tan (u + v) =

Tan (u – v) =

Tan u + Tan v1 - Tan u Tan v

Tan u - Tan v1 + 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

22

23

22

21

Simplify

46

42

426

Sum and Difference Formulas

127 Cos

4

3

127

)4

3

( Cos 127 Cos

4Sin

3Sin -

4 Cos

3 Cos

127 Cos

Sum and Difference Formulas

4Sin

3Sin -

4 Cos

3 Cos

127 Cos

21

22

23

22

462

Sum and Difference Formulas Evaluate the following functions.

a)

b)

o75 Cos

12Sin

426

426

Sum and Difference Formulas

o75 Cos = Cos 45º Cos 30º - Sin 45º Sin 30º

22

23

22

21

426

Sum and Difference Formulas

12

Sin

23

22

21

22

426

4Sin

3 Cos -

4 Cos

3Sin

Sum and Difference Formulas

Section 5.4

Sum and Difference Formulas Evaluate the following functions.

a)

b)

o105- Cos

125Tan

462

23

Sum and Difference Formulas

o105- Cos = Cos 150º Cos 45º + Sin 150º Sin 45º

23-

22

21

22

462

Sum and Difference Formulas

125Tan )

46(Tan

)4

Tan()6

(Tan -1

)4

Tan()6

(Tan

33 - 1

133

333

333

Sum and Difference Formulas

125Tan

3-3 33

33 33

6 3612

32

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

What formula is being used here?→ Cos (u – v)

Re-write the expression using the formula→ Cos (78º – 18º)

= Cos 60º = ½

= ½

Sum and Difference Formulas Use the sum and difference formulas to

evaluate the following:oooo 12Sin 42 Cos - 12 Cos42Sin a)

5Sin

7Sin -

5 Cos

7 Cos b)

o30Sin 21

)57

( Cos

3512 Cos

Sum and Difference Formulas Find the exact value of the Cos (u – v) using the

given information:

Sin u = Cos v = Both u and v and in quadrant III 25

7

54

When you are given 2 different criteria, you must draw 2 different triangles

u-7

25

-24- 3

5

- 4v

Sum and Difference Formulas

u-7

25

-24- 3

5

- 4v

Cos (u – v) = Cos u Cos v + Sin u Sin v

2524

54

25

7

53

12596

12521

125117

Sum and Difference Formulas Find the exact value of the trig functions given

the following information:

Tan u = Csc v =

and both u and v are in quadrant IV.

Find a) Sin (u + v) b) Sec (u – v)

c) Cot (u – v)

43

5

13

5663

6365

6556

Sum and Difference Formulas

u-3

5

4v

-513

12

Sin (u + v) = Sin u Cos v + Cos u Sin v

53

1312

54

13

5

6536

6520

6556

Sum and Difference Formulas

u-3

5

4v

-513

12

Sec (u - v) = Cos u Cos v + Sin u Sin v

54

1312

53

13

5

6548

6515

6563

1 ÷ Cos (u - v)Cos (u - v) =

6365

Sum and Difference Formulas

u-3

5

4v

-513

12

Cot (u - v) =

125

431

125

43

1 ÷ Tan (u - v)

Tan (u - v) = Tan vu Tan 1Tan v -u Tan

4815

4848

4820

4836

48634856

6356

5663

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 xTan u = 1 Cos v = x

→ Use this information to draw your triangles.

Sum and Difference FormulasTan u = 1 Cos v = x

u1

1

2

vx

1 21 x

Cos (u + v) = Cos u Cos v – Sin u Sin v

21

x2

1 21 x

2x

2

1 2x

21 2xx

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 xTan u = 2x Cos v = x

Sum and Difference Formulas

u 2x

1

241 x

vx

1 21 x

Sin (u – v) = Sin u Cos v – Cos u Sin v

241

2

x

x

x

241

1

x 21 x

2

2

41

2

x

x

2

2

41

1

x

x

2

22

41

12

x

xx

Tan u = 2x Cos v = 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 2xSin u = 3x Tan v = 2x

Sum and Difference Formulas

u 3x1

291 xv

1

x2

Cos (u + v) = Cos u Cos v – Sin u Sin v291 x 241

1

x x3

241

2

x

x

2

2

41

91

x

x

2

2

416x

x

2

22

41

691

x

xx

Sin u = 3x Tan v = 2x241 x

Sum and Difference Formulas So far, in this section we have:

a) Used sum and difference formulas to evaluate trig functions of different angles

b) Recognized sum and difference formulas to simplify expressions

c) Used criteria to draw triangles and apply formulas

d) 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 FormulasTan (π + x) = Tan x

Tan x Tan 1Tan x Tan

Tan x (0) 1

Tan x 0

1Tan x

Tan x

Sum and Difference FormulasSin (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

1 )4

-(x Cos - )4

(x Cos

4Sin Sin x

4 Cos x Cos

)4

Sin Sin x 4

Cos x (Cos -

4Sin Sin x

4 Cos x Cos

)4

Sin Sin x 4

Cos x Cos-

4Sin 2Sin x

1

Sum and Difference Formulas

1 )4

-(x Cos - )4

(x Cos

4Sin 2Sin x

1 1 Sin x 222

1 Sin x 2 2

1 - Sin x

x ,4

5 4

7

Sum and Difference Formulas

1 )3

(x Sin )3

(x Sin

3Sin x Cos

3 CosSin x

3

Sin x Cos 3

CosSin x

3 Cos2Sin x

1

Sum and Difference Formulas

1 Sin x 212

1 Sin x x

2

1 )3

(x Sin )3

(x Sin

3 Cos2Sin x

1