5.4 sum and difference formulas

5.4 Sum and Difference 5.4 Sum and Difference FormulasFormulas

Addition and Subtract of the Sine function

With the sine function the sign between the expressions stays the same.

vSinuCosvCosuSinvuSin

vSinuCosvCosuSinvuSin

Find the Sine of u + v

3045 vanduLet

4

26

4

2

4

63045

2

1

2

2

2

3

2

23045

304530453045

Sin

Sin

SinCosCosSinSin

Find the Sine of u - v

3045 vanduLet

4

26

4

2

4

63045

2

1

2

2

2

3

2

23045

304530453045

Sin

Sin

SinCosCosSinSin

Addition and Subtract of the Cosine function

With the cosine function the sign between the expressions change.

vSinuSinvCosuCosvuCos

vSinuSinvCosuCosvuCos

Find the Cosine of u + v

3045 vanduLet

4

26

4

2

4

63045

2

1

2

2

2

3

2

23045

304530453045

Cos

Cos

SinSinCosCosCos

Find the Cosine of u - v

3045 vanduLet

4

26

4

2

4

63045

2

1

2

2

2

3

2

23045

304530453045

Cos

Cos

SinSinCosCosCos

Addition and Subtract of the Tangent function

Both are Rational expressions

vu

vuvu

vu

vuvu

tantan1

tantantan

tantan1

tantantan

Find the Tangent of u + v

3045 vanduLet

33

11

33

13045tan

30tan45tan1

30tan45tan3045tan

Find the Tangent of u + v

3045 vanduLet

33

11

33

13045tan

30tan45tan1

30tan45tan3045tan

326

3612

33

33

33

33

33

33)3045tan(

Find the Tangent of u - v

3045 vanduLet

33

)1(1

33

13045tan

30tan45tan1

30tan45tan3045tan

tantan1

tantantan

vu

vuvu

Find the Tangent of u - v

3045 vanduLet

33

)1(1

33

13045tan

30tan45tan1

30tan45tan3045tan

tantan1

tantantan

vu

vuvu

326

3612

33

33

33

333045tan

33

333045tan

Simplify

Remember: sin(u-v)=sin u·cos v-cos u·sin v

36

7

36

7

36

7 SinCosCosSinSin

Simplify

Remember: sin(u-v)=sin u·cos v-cos u·sin v

2

3

2

3

2

1

2

1

36

7

36

7

36

7

36

7

Sin

SinCosCosSinSin

Simplify

Remember: sin(u-v)=sin u·cos v-cos u·sin v

2

1

4

2

4

3

4

1

2

3

2

3

2

1

2

1

36

7

36

7

36

7

36

7

Sin

SinCosCosSinSin

Break up the angle and find Sin, Cos and Tan

Leave as a Rational Expression

12

11

Break up the angle and find Sin, Cos and Tan

Leave as a Rational Expression

64

3

12

11

Break up the angle and find Sin, Cos and Tan

Leave as a Rational Expression

64

3

64

3

64

3sin

SinCosCosSin

Break up the angle and find Sin, Cos and Tan

Leave as a Rational Expression

2

1

2

2

2

3

2

2

64

3

64

3

64

3sin

SinCosCosSin

Break up the angle and find Sin, Cos and Tan

Leave as a Rational Expression

4

26

4

2

4

6

2

1

2

2

2

3

2

2

64

3

64

3

64

3sin

SinCosCosSin

Break up the angle and find Sin, Cos and Tan

Leave as a Rational Expression

4

26

4

2

4

6

2

1

2

2

2

3

2

2

64

3

64

3

64

3cos

SinSinCosCos

Break up the angle and find Sin, Cos and Tan

Tan can be done two ways.

32

31

33

33

11

33

1

6tan

43

tan1

6tan

43

tan

64

3tan

3226

26

426

426

643

643

64

3

Cos

SinTan

Sin u = 5/13

What is Cos u ?

Sin u = 5/13

What is Cos u ?

13

12;

169

144cos

169

144

13

51cos

1cos13

5

1cossin

2

22

22

22

uu

u

u

uu

Find cos(arc cos x – arc tan x)

Let u = arc cos x

v = arc tan x

u

x

121 x

v

x12 x

1

Find cos(arc cos x – arc tan x)

u

x

121 x

v

x12 x1

1cos

1sin

cos

1sin

2

2

2

xv

x

xv

xu

xu

1

Find cos(arc cos x – arc tan x)

1

1cos

1sin

cos

1sin

2

2

2

xv

x

xv

xu

xu

1

1

1

1

1

11

1

1

sinsincoscoscos

2

2

2

2

2

2

2

2

x

xx

x

x

x

x

x

xx

xx

vuvuvu

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