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Trigonometri SMA Kelas XI IPA Sem 1

Jul 11, 2015

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Page 1: Trigonometri SMA Kelas XI IPA Sem 1

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Page 2: Trigonometri SMA Kelas XI IPA Sem 1

Setelah menyaksikan tayangan ini anda dapat

Menyelesaikansoal-soal yang berkaitan

dengan jumlah dan selisih sudutserta sudut rangkap

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Page 3: Trigonometri SMA Kelas XI IPA Sem 1

Rumusjumlah dan selisih dua sudut

sin(α + β)

= sinα.cosβ + cosα.sinβ sin(α - β)

= sinα.cosβ - cosα.sinβ

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Page 4: Trigonometri SMA Kelas XI IPA Sem 1

1. Sin 75o = …. Bahasan: sin(α + β) = sinα.cosβ + cosα.sinβ sin750 = sin(450 + 300)

= sin450cos300 + cos450sin300

= ½√2.½√3 + ½√2.½ = ¼√6 + ¼√2 = ¼√2(√2 + 1)

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Page 5: Trigonometri SMA Kelas XI IPA Sem 1

2. Diketahui sin A = cos B =

A dan B adalah sudut-sudut lancip

sin(A – B) =….

Bahasan:

sin(A – B)= sinAcosB – cosAsinB

sinA =

cosA =

5

3

25

7

? ?

A

5

33

5

45

4B

cos B =

sin B =25

7

7

252425

24

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Page 6: Trigonometri SMA Kelas XI IPA Sem 1

sin A = → cos A = cos B = → sin B =

sin(A – B) =….

= sinAcosB – cosAsinB

= x - x

=

=

5

3

25

7 5

4

25

24

5

3

25

7

5

4

25

24

125

96

125

21 −

5

3

125

75 −=−

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Page 7: Trigonometri SMA Kelas XI IPA Sem 1

Rumus

jumlah dan selisih dua sudut cos(α + β)

= cosαcosβ - sinαsinβ

cos(α - β)

= cosαcosβ + sinαsinβ

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Page 8: Trigonometri SMA Kelas XI IPA Sem 1

1. Bahasan: cosαcosβ + sinαsinβ = cos(α - β)

=

= =

....sinsincoscos 2813

75

2813

75 =+ ππππ

=+ 2813

75

2813

75 sinsincoscos ππππ )cos( 28

137

5 ππ −

)cos( 287 π

)cos( 4π

22

1

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Page 9: Trigonometri SMA Kelas XI IPA Sem 1

2. a. –sina.sinb b. cosa.cosb c. sina.sinb d. 1 – tana.tanb e. 1 + tana.tanb

....bcos.acos

)bacos( =+

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Page 10: Trigonometri SMA Kelas XI IPA Sem 1

=

= 1 – tana.tanb → jawab d

=−bcos.acos

)bacos(

bcos.acos

bsin.asinbcos.acos +

bcos.acos

bsin.asin

bcos.acos

bcos.acos +

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Page 11: Trigonometri SMA Kelas XI IPA Sem 1

3. Tentukan

nilai cos56° + sin56°.tan28°

Bahasan: cos56° + sin56°.tan28°

= cos56° + sin56°.

= cos56° +

0

0

28cos

28sin

0

00

28cos

28sin.56sin

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Page 12: Trigonometri SMA Kelas XI IPA Sem 1

= cos56° +

=

=

=

Jadi,

Nilai cos56° + sin56°.tan28° = 1

0

0000

28cos

28sin.56sin28cos56cos +

0

0

28cos

28sin = 1

0

00

28cos

)2856cos( −

0

00

28cos

28sin.56sin

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Page 13: Trigonometri SMA Kelas XI IPA Sem 1

4. Pada suatu segitiga siku-siku ABC berlaku cosA.cosB = ½. Maka cos(A – B) =…. Bahasan: ∆ siku-siku ABC; cosA.cosB = ½ maka ΔABC siku-siku di C →∠C = 90° A + B + C = 180° → A + B = 90°

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Page 14: Trigonometri SMA Kelas XI IPA Sem 1

A + B + C = 180° → A + B = 90° A = 90° – B → B = 90° – Acos(A – B) = cosA.cosB + sinA.sinB = ½ + sin(90 – B).sin(90-A) = ½ + cosB.cosA = ½ + ½ = 1Jadi cos(A – B) = 1

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Page 15: Trigonometri SMA Kelas XI IPA Sem 1

Rumusjumlah dan selisih dua sudut

tan(α + β) =

tan(α - β) =

βαβα

tan.tan1

tantan

−+

βαβα

tan.tan1

tantan

+−

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Page 16: Trigonometri SMA Kelas XI IPA Sem 1

1. tan 105° = …. Bahasan:

tan105° = tan(60° + 45°)

oo

oo

45tan.60tan1

45tan60tan

−+=

1.31

13

−+=

x31

31

−+=

31

31

++

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Page 17: Trigonometri SMA Kelas XI IPA Sem 1

tan 105° = x

=

=

=

= -2 - √3

31

31

−+

31

31

++

31

)31( 2

−+

31

3321

−++

2

324

−+

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Page 18: Trigonometri SMA Kelas XI IPA Sem 1

2. Diketahui A + B = 135° dan tan B = ½. Nilai tan A= …. Bahasan: A + B = 135°

tan(A + B) = tan 135° = -1 = -1

Btan.Atan1

BtanAtan

−+

21

21

.Atan1

Atan

−+

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Page 19: Trigonometri SMA Kelas XI IPA Sem 1

= -1 tan A + ½= -1 + ½tan A tan A - ½tan A = -1 - ½ ½tan p = -1½ Jadi, tan p = -3

21

21

.Atan1

Atan

−+

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Page 20: Trigonometri SMA Kelas XI IPA Sem 1

3. Jika tan q = ½ dan p – q = ¼π maka tan p = …. Bahasan: p – q = ¼π tan(p – q) = tan ¼π = 1 = 1

qtan.ptan1

qtanptan

+−

21

21

.ptan1

ptan

+−

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Page 21: Trigonometri SMA Kelas XI IPA Sem 1

= 1 tan p - ½ = 1 + ½tan p tan p - ½tan p = 1 + ½ ½tan p = 1½ Jadi, tan p = 3

21

21

.ptan1

ptan

+−

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Page 22: Trigonometri SMA Kelas XI IPA Sem 1

Rumus Sudut Rangkap

sin2a = 2 sina.cosa

contoh: 1. sin10° = 2sin5°.cos5°

2. sin6P = 2sin3P.cos3P

3. sin t = 2sin½t.cos½t

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Page 23: Trigonometri SMA Kelas XI IPA Sem 1

1.Diketahui cosα =

Nilai sin 2α =….

Bahasan:

cos α =

sinα =

α4

3

5

3

55

3

5

4

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Page 24: Trigonometri SMA Kelas XI IPA Sem 1

cos α =

sinα =

Jadi sin2α = 2sinα.cosα = 2. x =

α4

3

55

3

5

4

5

4

5

3

25

24

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Page 25: Trigonometri SMA Kelas XI IPA Sem 1

2. Jika tan A = ½ maka sin 2A =….

Bahasan:

tan A = ½ sinA = dan cosA =

sin2A = 2 sinA.cosA

= 2 x x =

A1

2

512 22 =+5

1

5

2

5

1

5

2

5

4

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Page 26: Trigonometri SMA Kelas XI IPA Sem 1

3. Jika sinx – cosx = p maka harga sin 2x =….

Bahasan: sinx – cosx = p

(sinx – cosx)2 = p2

sin2x – 2sinx.cosx + cos2x = p2

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Page 27: Trigonometri SMA Kelas XI IPA Sem 1

sin2x – 2sinx.cosx + cos2x = p2

sin2x + cos2x – 2sinx.cosx = p2

1 – sin2x = p2

1 – p2 = sin2x

Jadi, harga sin2x = 1 – p2

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Page 28: Trigonometri SMA Kelas XI IPA Sem 1

4. Diketahui A adalah sudut lancip

dan cos½A =

Nilai sin A = ….

Bahasan:

cos½A =

dengan phytagoras

t2 = 2x – (x + 1)

t = √x - 1

½A√x+ 1

x2

1x +

x2

1x +

√2xt = √x - 1

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Page 29: Trigonometri SMA Kelas XI IPA Sem 1

cos½A = → sin½A =

sinA = 2sin½A.cos½A

= 2 x x

= Jadi, sin2x =

½A√x+ 1

x2

1x +

√2xt = √x - 1

x2

1x −

x2

1x −

x2

1x +

x

1x 2 −x

1x 2 −

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Page 30: Trigonometri SMA Kelas XI IPA Sem 1

Rumus Sudut Rangkap

cos 2a = cos2a – sin2a

= 2cos2a – 1

= 1 – 2sin2a

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Page 31: Trigonometri SMA Kelas XI IPA Sem 1

1. Diketahui cosα =

maka cos 2α =….

Bahasan:

cos2α = 2cos2α - 1

= 2( )2 – 1

= - 1

= -

3

1

3

1

9

2

9

7

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Page 32: Trigonometri SMA Kelas XI IPA Sem 1

2. Diketahui sinx = ½ maka cos 2x =….

Bahasan:

cos2x = 1 – 2sin2x

= 1 – 2(½)2

= 1 – ½ = ½

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Page 33: Trigonometri SMA Kelas XI IPA Sem 1

3. Diketahui tan p = ½ maka cos 2p =….

Bahasan:

tan p = ½ →

cos2p = 1 – 2sin2p

= 1 – 2( )2

= 1 –

=

p1

2

√5

sin p = 5

1

5

1

5

2

5

3

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Page 34: Trigonometri SMA Kelas XI IPA Sem 1

4. Diketahui sudut lancip A

dengan cos 2A =

Nilai tan A = ….

Bahasan: • cos 2A = 1 – 2sin2A

= 1 – 2sin2A

2sin2A = 1 – =

3

1

3

1

3

1

3

2

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Page 35: Trigonometri SMA Kelas XI IPA Sem 1

• cos 2A = 2cos2A – 1

= 2cos2A – 1

2cos2A = + 1 =

tan2A = =

tan2A = ½ A lancip → Jadi, tan A = ½√2

3

1

3

1

3

4

2sin2A 2cos2A

34

32

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Page 36: Trigonometri SMA Kelas XI IPA Sem 1

5. Diketahui A adalah sudut lancip

dan cos½A =

Nilai sin A adalah…. Bahasan: cos A = 2cos2½A – 1

= 2 - 1

= 2 - 1

=

x2

1x +

2

x2

1x

+

+

x2

1x

x

1

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Page 37: Trigonometri SMA Kelas XI IPA Sem 1

cos x = 2 - 1

cos x =

cos x = →

Jadi, nilai sin x =

x1

x√x2 – 1

x

1x 2 −

x

1

+

x2

1x

x2

x22x2 −+

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Page 38: Trigonometri SMA Kelas XI IPA Sem 1

6. Buktikan:

Bahasan:

atanasin

acos121=−

=−asin

acos1aa

a

21

21

212

cossin2

)sin21(1 −−

acosasin2

asin2

21

21

212

=

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Page 39: Trigonometri SMA Kelas XI IPA Sem 1

=−asin

acos1

acosasin2

asin2

21

21

212

acos

asin

21

21

=

atan 21=

Terbukti : =−asin

acos1atan 2

1

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Page 40: Trigonometri SMA Kelas XI IPA Sem 1

Rumus Sudut Rangkap

tan 2a =

Contoh: 1. tan 20° =

2. tan 10x =

atan1

atan22−

02

0

10tan1

10tan2

x5tan1

x5tan22−

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Page 41: Trigonometri SMA Kelas XI IPA Sem 1

1. Jika tan A = 3 maka tan 2A =….

Bahasan:

tan 2A =

=

= =

Atan1

Atan.22−

231

3.2

8

6

− 4

3−

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Page 42: Trigonometri SMA Kelas XI IPA Sem 1

2. Jika cos x =

maka tan 2x =….

Bahasan:

tan 2x =

=

=

Atan1

Atan.22−

( ) 2

512

512

1

.2

25144

524

1 −

13

5

x5

1312

tan x =5

12

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Page 43: Trigonometri SMA Kelas XI IPA Sem 1

tan 2x =

=

=

Jadi, tan 2x =

25144

524

1 −

25144 25

524

119

5.24

− 119

120−

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Page 44: Trigonometri SMA Kelas XI IPA Sem 1

SELAMAT BELAJAR

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