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Topic : - THE MULTIPLICATION OF SINE AND COSINE FORMULAS

- THE SUM AND DIFFERENCE OF SINE AND COSINE FORMULAS TIME : 4 X 45 minutes

STANDARD COMPETENCY:

2. To derive the formulas of trigonometry and its applications.

BASIC COMPETENCY:

2.2 To determine the trigonometric formulas of the Sum and Difference of Sine and Cosine.

In this chapter, you will learn:

• To prove the formulas of the multiplication of sine and cosine.

• To use the formulas of the multiplication of sine and cosine.

• To prove the formulas of the sum and difference of sine and cosine.

• To use the formulas of the sum and difference of sine and cosine.

C. The Multiplications of sine and cosine We have proved that 1. ( )sin sin cos cos sinα β α β α β+ = ⋅ + ⋅ and

2. ( )sin sin .cos cos .sinα β α β α β− = −

If we add them, we get

( )sin sin cos cos sinα β α β α β+ = ⋅ + ⋅

( )sin sin cos cos sinα β α β α β− = ⋅ − ⋅ +

( ) ( ) .................................sinsin =−++ βαβα

Then we have

( ) ( )βαβα −++= sinsin.................................

Worksheet Worksheet Worksheet Worksheet 4444thththth

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If we subtract them, we get

( )sin sin cos cos sinα β α β α β+ = ⋅ + ⋅

( )sin sin cos cos sinα β α β α β− = ⋅ − ⋅

( ) ( )sin sin .................................α β α β+ − − =

Then we have

( ) ( )...................................... sin sinα β α β= + − −

We also have already proved: 3. ( )cos cos cos sin sinα β α β α β+ = ⋅ − ⋅ and

4. ( )cos cos cos sin sinα β α β α β− = ⋅ + ⋅

If we add them, we get

( )cos cos cos sin sinα β α β α β+ = ⋅ − ⋅

( )cos cos cos sin sinα β α β α β− = ⋅ + ⋅ +

( ) ( ) .................................coscos =−++ βαβα

Then we have

( ) ( )βαβα −++= coscos.................................

If we subtract them, we get

( )cos cos cos sin sinα β α β α β+ = ⋅ − ⋅

( )cos cos cos sin sinα β α β α β− = ⋅ + ⋅

( ) ( )cos cos .................................α β α β+ − − =

Then we have ( ) ( )βαβα −−+= coscos......................................

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Example 23 Write in the addition form: a. 0 02 cos80 sin 50 b. 2 sin cos3a a

Solution

a. 0 02 cos80 sin 50 = ( ) ( )00 sinsin KKKKKKKK −−+

= 00 sinsin KKKK − = KKKK −0sin

b. 2 sin cos3a a = ( ) ( )KKKKKKKK −++ sinsin

= KKKK sinsin + = KKKK −

Example 24 Write in the addition form:

a. 0 02 cos75 cos15 b. 3

sin sin8 8

π π ⋅

Solution

a. 0 02 cos75 cos15 = ( ) ( )00 coscos KKKKKKKK −++

= 00 coscos KKKK + = KKKKKK =+

b. 3

sin sin8 8

π π ⋅

= 1 3

2 sin sin2 8 8

π π − − ⋅

= 1 3 3

cos cos2 8 8

π π − + − −

KK KK

= { }KKKK coscos2

1 −−

= ( )KKKK −−2

1= KKK

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Exercise 4 1. Write each term below in the sum and difference of two angles.

a. 2 sin 3 cos3A A f. 2 sin cos3α α

b. 6 sin 6 sin 2α α g. 2 sin 2 sin 4β β

c. 4 sin 4 sinα α h. cos 4 cos 2α α

d. 3 1

sin sin2 2

α α⋅ i. 5

5 cos sin2 2

α α⋅

e. sin 6 cos 4α α

2. Calculate without calculator: a. 0 02 sin 75 cos15 d. 2 0 2 0sin 195 sin 75

b. 3

4 sin cos8 8

π π e. cos cos

8 8

π π

c. 0 0sin105 sin15 f. 2 211 54 cos cos

4 24π π⋅

3. Write as sum and difference of sine and cosine, and simplify as possible.

a. 2 cos( ) cos( )x y x y+ ⋅ − e. 0 0cos50 sin 30

b. 2 sin(2 ) sin(2 )x y x y+ ⋅ − f. 0 0sin 22 cos 66

c. 1 12 22 cos ( ) cos ( )a b a b+ ⋅ − g. 0 0sin130 cos 20

d. 2 sin( ) sin( )A B C A B C+ − ⋅ − + h. 0 0cos 25 sin 75

e. 2 sin( ) cos( )A B A B+ ⋅ − i. 1 12 22 sin ( ) cos ( )x y x y+ ⋅ −

f. 1 12 22 sin ( ) sin ( )θ π θ π+ ⋅ − j. 1 1

4 42 cos( ) cos( )x xπ π+ ⋅ −

4. Prove the identity below.

a. 2 22 cos( ) cos( ) 1 cos cosα β α β α β+ ⋅ − + = +

b. cos5 sin cos sin 3 cos 2 sin 4 0θ θ θ θ θ θ⋅ + ⋅ − ⋅ =

c. 3

2 cos cos 1 sin 24 4

π πθ θ θ + ⋅ − + =

d. 1

sin sin cos 24 4 2

π πθ θ θ + ⋅ − =

e. sin 7 sin 5 sin 3 sin 4 cos cos 2 sin 4θ θ θ θ θ θ θ+ + + = ⋅ ⋅

5. Show that 0 0 08 sin 20 sin 40 sin 80 3= and without calculator calculate: 0 0 08 sin 70 sin 50 sin10 .

6. Prove this identity below.

a. ( ) ( ) bababa 22 sinsinsin.sin −=−+

b. ( ) ( ) bababa 22 sincoscos.cos −=−+

c. ( ) ( ) ( ) ( ) ( )0 0 0 0sin 45 cos 45 cos 45 sin 45 cosA B A B A B+ − + + − = −

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d. 3 32 sin cos 2 sin cos sin 2x x x x x+ =

e. 21 cos5 cos3 sin 5 sin 3 2 sinθ θ θ θ θ− − =

f. ( )22 cos sin 3 sin 2 2 cos 1θ θ θ θ= +

7. Without calculator, prove that 0 0 0 0 0 0 1sin 52 sin 68 sin 47 cos77 cos65 cos81

2− − =

8. Prove:

a. 3

2 sin sin cos 24 4

π πα α α + + =

b. 0 0 0 0 04 sin18 cos36 sin 54 1 2 sin18 cos 36= + − 9. Prove:

a. 2 sin 3 sin 4 2 cos5 cos 2 cos3 cosx x x x x x+ − =

b. ( )( )sin 3 cos sin 1 2 sin 2 cos3B B B B B+ + − =

10. Prove 3 5 3 3 71

2 2 2 2 2 22 sin cos 2 sin cos 2 sin cos sin 4 sin 5θ θ θ θ θ θ θ θ⋅ + ⋅ + ⋅ = +

E. The Sum and Difference Formula of Sine and Cosine

We have proved that: 1. ( ) ( )2 sin cos sin sinα β α β α β= + + −

2. 2 cos sin .................. .......................α β = −

3. 2 cos cos .................. .......................α β = +

4. 2 sin sin .................. .......................α β− = −

If x=+ βα and y=− βα , then we have

x=+ βα and x=+ βα

y=− βα + y=− βα -

KKKK +=α2 KKKK −=β2

KKKK=α KKKK=β

We substitute to the formula (1) ( ) ( )2 sin cos sin sinα β α β α β= + + − , then we get

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( ) ( )1 12 sin cos sin sin

2 2x y x y x y+ − = +

We can prove the sum and difference formula of Sine and Cosine:

( ) ( )1 1sin sin 2 sin cos

2 2x y x y x y+ = + ⋅ −

................................................sinsin =− yx

................................................coscos =+ yx

................................................coscos =− yx Example 25 Write xx 2sin6sin + in the multiplication form:

Solution

xx 2sin6sin + = ( ) ( )1 12 22 sin cos+ ⋅ −KK KK KK KK

= ( ) ( )1 12 22 sin cos⋅KK KK

= 2 sin .cosKK KK Example 26 Simplify 00 15cos105cos − !

Solution

00 15cos105cos − = ( ) ( )1 1

2 22 sin sin− + ⋅ −KK KK KK KK

= ( ) ( )1 1

2 22 sin sin− ⋅KK KK

= 2 sin .sin− KK KK

= ( ) ( )2− ⋅KK KK

= ………

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Example 27 Calculate 00 15cos195cos + !

Solution

00 15cos195cos + = ( ) ( )1 1

2 22 cos cos+ −KK KK KK KK

= ( ) ( )1 1

2 22 cos cosKK KK

= 2 cos .cosKK KK = ( ) ( )2 ⋅KK KK

= ……… Exercise 4 1. Write each term below in the multiplication form and simplify as possible.

a. xx sin5sin + g. AA 2cos8cos + b. αα cos3cos + h. θθ cos3cos − c. ββ sin5sin − i. xx 3sin7sin +

d. αα sin3sin − j. xx 2sin6sin − e. θθ 3cos5cos + k. ( ) xhx sinsin −+

f. ( ) xhx coscos −+ l. ( ) ( )βαβα −++ sinsin

2. Simplify: a. 00 18sin42sin + e. ( ) ( )yxyx −++ 2sin2sin

b. 00 55cos125cos + f. ( ) xhx coscos −+

c. 00 20cos200cos − g. ( ) ( )θπθπ ++− 21

21 sinsin

d. 00 80sin100sin − h. ( ) ( )θπθπ 2sin2sin 21

21 −−+

3. Calculate without calculator.

a. 00 15cos75cos − d. 00 15cos105cos +

b. 00 15sin105sin − e.

−

12

5cos

12cos

ππ

c. 00

00

75sin15sin

75cos15cos

−−

f. 00

00

15sin105sin

195cos225cos

−−

4. Prove these identities below.

a. θθθθθ

tan5cos7cos

5sin7sin =+−

d.

−=+−

2tan

coscos

sinsin φθφθφθ

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b. θθθθθ

tan4sin6sin

4cos6cos −=+−

e. ( )βαβαβα −−=

+−

tan2sin2sin

2cos2cos

c. θθθθθ

2tan3coscos

3sinsin =++

f. AAA

AA

2

1cos

2coscos

2sinsin =−+

5. Calculate without calculator.

a. 000 35cos85cos155cos ++

b.

−

+

18

7sin

18sin

18

3sin

πππ

6. Prove:

a. xxx

xx3tan

2cos4cos

2sin4sin =++

b. cos3 cos5

2 sin 2sin 3 sin

x xx

x x

− =−

7. If θθ sin3sin +=x and θθ cos3cos +=y , then prove:

a. ( )2 cos sin 2 cos 2x y θ θ θ+ = +

b. tan 2x

yθ=

c. 2 2 2 2 cos 2x y θ+ = +

8. If p=+ βα sinsin and q=+ βα coscos , then prove:

a. ( ) ( ) ( )1 1 12 2 22 cos sin cosp q α β α β α β + = − + + +

b. ( )12tanp q α β= +

c. ( ){ } ( )2 2 2 122 1 cos 4 cosp q α β α β+ = + − = −

9. Prove the identity:

a. ( )sin sin 2 sin 3 sin 2 2 cos 1A A A A A+ + = +

b. 2cos 2 cos3 cos5 4 cos3 cosθ θ θ θ θ+ + =

c. ααααααα

2tan3cos2coscos

3sin2sinsin =++++

10. Prove:

a. ( )xxxxx 6cos4coscot4sin6sin −=+

b. ( ) ( )yx

yxyxyx

coscos

coscostan.tan 2

121

+−=−+

c. sin 2 sin 4 sin 6 4 cos cos 2 sin 3a a a a a a+ + = ⋅ ⋅

d. ( ) xxxxx 3sinsin3coscos2tan +=+

11. If A , B and C are the angles of the triangle ABC , then prove:

a. ( )sin sin 2 sin cosB A C A C+ − =

b. sin 2 sin 2 sin 2 4 sin sin sinA B C A B C+ + = ⋅ ⋅

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c. 1 cos 2 cos 2 cos 2 4 sin sin cosC A B A B C+ − − = ⋅ ⋅ n

12. Prove that:

a. ( )sin

tan tancos cos

x yx y

x y

++ =

⋅

b. ( )

yx

yxyx

cos.cos

sintantan

−=−

13. Prove that ( )sin

tan tancos cos

α βα β

α β−

= −⋅

14. If ( ) ( )2 cos cosx y x y+ = − , then prove 3

1tan.tan =yx

15. Prove that 0 0 0 0 0 0 0 0tan 5 tan 40 tan10 tan 35 tan10 tan 35 tan 5 tan 40− = + − − 16. Calculate without calculator:

a. 0 0 0csc 10 csc 50 csc70+ −

b. 0 0 0 0 0 0 0cos12 cos 24 cos36 cos 48 cos60 cos72 cos84 17. Prove that 5 3sin 5 16 sin 20 sin 5 sinx x x x= − + . (Use: xxx 235 += or xxx += 45 ) 18. If A , B and C are the angles of the triangle ABC , then prove

1 1 12 2 2sin sin sin 4 cos cos cosA B C A B C+ + = ⋅ ⋅

19. Calculate: 0000 3tan39tan33tan69tan −−+ . 20. Calculate: 0 0 0 0 0 0sin 52 sin 68 sin13 sin 47 sin 9 sin 25− − .