Topic : - THE MULTIPLICATION OF SINE AND COSINE … cos sin3 sin2 2 cos 1θ θ θ θ= +(2) 7....
Transcript of Topic : - THE MULTIPLICATION OF SINE AND COSINE … cos sin3 sin2 2 cos 1θ θ θ θ= +(2) 7....
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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− − .