Download - Trigonometri ok
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MENERAPKAN PERBANDINGAN,FUNGSI,PERSAMAAN,DAN
IDENTITAS TRIGONOMETRI DALAM PEMECAHAN MASALAH
STANDAR KOMPETENSI
Trigonometri
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KOMPETENSI DASAR1. MENENTUKAN NILAI PERBANDINGAN
TRIGONOMETRI SUATU SUDUT
2. MENGKONVERSI KOORDINAT KARTESIUS DAN KUTUB
3. MENERAPKAN ATURAN SINUS DAN KOSINUS
4. MENENTUKAN LUAS SUATU SEGITIGA
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1. MENENTUKAN NILAI PERBANDINGAN TRIGONOMETRI SUATU SUDUT
a. PERBANDINGAN TRIGONOMETRI PADA BIDANG SEGITIGA SIKU-SIKU
b. PANJANG SISI DAN BESAR SUDUT SEGITIGA SIKU-SIKU
c. PERBANDINGAN TRIGONOMETRI DI BERBAGAI KUADRAN
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2. MENGKONVERSI KOORDINAT KARTESIUS DAN KUTUB
a. Koordinat kartesius dan kutub
b. Konversi koordinat kartesius dan kutub
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3. MENERAPKAN ATURAN SINUS DAN KOSINUS
a. Aturan sinus dan kosinus
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4. MENENTUKAN LUAS SUATU SEGITIGA
a. Luas segitiga
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pengertian PERBANDINGAN TRIGONOMETRI
PERBANDINGAN YANG TERDAPAT PADA SEGITIGA SIKU-SIKU YANG TIDAK DIBATASI OLEH SUMBU KARTESIUS
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PANJANG SISI DAN BESAR SUDUT SEGITIGA SIKU-SIKU
αA
C
B
ab
c
1. Sinus α =
2. Cosinus α =
3. Tangan α =
b
a
AC
BC
miringsisi
Adgnberhadapanyangsisi ==∠
c
a
AB
BC
Adgnanberdampingyangsisi
Adgnberhadapanyangsisi ==∠
∠
b
c
AC
AB
miringsisi
Adgnanberdampingyangsisi ==∠
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PERHATIKAN PADA BANGUN YANG LAIN
Perbandingan Trigonometri pada
bangun yang lain :
P Q
R
Cos Q =
Sin Q =
Tg Q =
Sin R =
Cos R =
Tg R =
QR
PR
QR
PQ
PQ
PR
QR
PQ
QR
PR
PR
PQ
KEMBALI KE ….
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PERHATIKAN CONTOH BERIKUT :
Perhatikan gambar
10 cm
AB
C
300
No. 1
a. Tentukanlah panjang AB
b. Tentukanlah panjang BC
Jawab
Cos 300 =
Sin 300 =……… ?
Rumus fungsi yang mana yang kita gunakan ?
AC
AB⇒ 030Cos)AC(AB =
030Cos).10(AB =3
2
1).10(AB =
⇒
⇒
⇒ 35AB =
Silahkan anda coba
Catatan : Nilai Sin/Cos dapat dilihat pada tabel
AC
ABCoba anda cari BCDengan Menggunakan fungsi apa ?
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PERHATIKAN CONTOH YANG LAINNo. 2
Jika diketahui segitiga ABC siku-siku di ∠ C, panjang AB = 25 cm, AC =
9 cm
Tentukanlah :
a. Besar ∠ A
b. B Besar ∠ B
Jawab :
Fungsi Trigono yang mana yang kita pergunakan ?
cos A = …. Karena yang diketahui AC dan AB
AB
ACACos = ⇒ 6,0
5
3
25
9 ===ACos ⇒ 6,0CosA =
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Lanjutkan ke
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PERBANDINGAN TRIGONOMETRI PADA SEGITIGA DALAM SUMBU
KARTESIUS
r
x
miringsisi
Adgnberhadapanyangsisi =∠
r
y
miringsisi
Adgnanberdampingyangsisi =∠
Sb y
Sb x
yr
x
1. Sinus α =
2. Cosinus α =
3. Tangan α = x
y
Adgnanberdampingyangsisi
Adgnberhadapanyangsisi =∠
∠
LANJUTKAN KE…
α
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SUDUT ISTIMEWAUntuk ∠ 300 dan ∠ 600
A B
C
600
300
2
1
3
Sin 300 =
Cos 300 =
Tg 300 =
Sin 600 =
Cos 600 =
Tg 600 =
2
1
AC
AB =
32
1
2
3
AC
BC ==
33
1
3
1
BC
AB ==
32
1
2
3
AC
BC ==
2
1=AC
AB
1
3
AB
BC =
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SUDUT ISTIMEWA
Untuk ∠ 450
Sin 450 =
Cos 450 =
Tg 450 =
450
450
AB
C
22
1
2
1
AC
BC ==
22
1
2
1
AC
AB ==
11
1
AB
BC ==1
12
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SUDUT ISTIMEWA
Untuk ∠ 00
X=r
Sb. : y
Sb.: x
Sin 00 =
Cos 00 =
Tg 00 =
0r
0
r
y ==
1r
r
r
x ==
0x
0
x
y ==
Catatan :
X = r
Y = 0
Y=0
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SUDUT ISTIMEWA
Untuk ∠ 900
Sin 900 =
Sin 900 =
Cos 900 =
y = r
X = 0
1r
r
r
y ==
0r
0
r
x ==
∞==0
y
x
y
Catatan :
X = 0
Y = r
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KESIMPULAN SUDUT ISTIMEWA
22
122
1
33
1
α 0O 30O 45O 60O 90O
Sin 0 1
Cos 1 0
Tg 0 1 ∞
Ctg ∞ 1 0
2
12
2
12
2
1
2
1
33
13
3
LANJUTKAN KE….
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SUDUT ISTIMEWA
• DIPEROLEH DARI
Perbandingan trigonometri sisi-sisi segitiga siku-siku
Sudut Istimewa segitiga siku-siku yaitu :
1. 00
2. 30o
3. 450
4. 60o
5. 90o
LANJUTKAN KE..
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PERBANDINGAN TRIGONOMETRI DI BERBAGAI KUADRAN
00 18090 << α 00 900 << α
00 270180 << α00 360270 << α
Sudut di Kuadran I = α Sin bernilai (+) Cos bernilai (+) Τan bernilai (+) Sudut di Kuadran II = β = (180 - α)Hanya Sin bernilai (+)
Sudut di Kuadran III =γ =(180 +α )Hanya Tan bernilai (+)
Sudut di Kuadran IV =θ =( 360 -α)Hanya Cos bernilai (+)
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KOORDINAT KUTUB DAN KARTESIUS
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KOORDINAT KUTUB
θ
r θ)B(r,
Koordinat Kutub
B(r,θ)
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KOORDINAT KARTESIUS
Koordinat kartesius A (x,y)y)A(x,
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MENGUBAH KOORDINAT KUTUB MENJADI KOORDINAT KARTESIUS
Koordinat kutub B(r,θ)
Dari diperoleh x = r . cos θ
sedangkan diperoleh y = r . sin θ
Sehingga didapat Koordinat kartesius B(x,y) = (r.Cosθ , r.Sinθ)
Cosθr
x =
Sinθr
y =
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MENGUBAH KOORDINAT KARTESIUS MENJADI KOORDINAT KUTUB
Koordinat kartesius A (x,y)
22 yxr +=
x
yTanθ =
x
yarc.Tanθ =
Sehingga koordinat kutub A (r,θ)
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ATURAN SINUS DAN KOSINUS
ATURAN SINUS
ATURAN KOSINUS
SinCc
SinBb
SinAa ==
2bcCosA2c2b2a −+=2acCosB2c2a2b −+=
2abCosC2b2a2c −+=
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KOMPETENSI DASAR 3KOMPETENSI DASAR 3
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ATURAN SINUS
SinCc
SinBb
SinAa ==
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Bukti :
SinΑb
CD =
aSinBCD =b.SinACD =
SinBa
CD =
aSinBbSinA =
SinB
b
SinA
a =
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CONTOH SOAL :CONTOH SOAL :
Pada segitiga ABC, diketahui
c = 6, sudut B = 600 dan sudut C = 450.
Tentukan panjang b !
0
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PENYELESAIAN :PENYELESAIAN :
2
6
3
45
6
60
21
21
00
=
=
=
bSinSin
bSinC
c
SinB
b
632
66
2
2
2
36
2
63
21
21
==
•=
×=
b
b
b
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ATURAN KOSINUS
2bcCosA2c2b2a −+=
2acCosB2c2a2b −+=
2abCosC2b2a2c −+=
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CONTOH SOAL :
Pada segitiga ABC, diketahui
a = 6, b = 4 dan sudut C = 1200 Tentukan panjang c
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PENYELESAIAN :
c2 = a2 + b2 – 2.a.b.cos Cc2 = (6)2 + (4)2 – 2.(6).(4).cos 1200
c2 = 36 + 16 – 2.(6).(4).( – ½ )c2 = 52 + 24 c2 = 76 c =√76 = 2√19