Trigonometry Tutorial * Right Triangles * Tangents * Small Angle Approximation * Triangulation
ADVANCED TRIGONOMETRY page 126
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Transcript of ADVANCED TRIGONOMETRY page 126
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ADVANCED
TRIGONOMETRY
page 126
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Kompetensi Dasar
2.1. Menggunakan rumus sinus dan kosinus jumlah dua sudut, selisih dua sudut, dan sudut ganda untuk menghitung sinus dan kosinus sudut tertentu
2.2. Menurunkan rumus jumlah dan selisih sinus dan cosinus.
2.3. Menggunakan rumus jumlah dan selisih sinus dan cosinus.
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TRIGONOMETRY 1
2.1. Menggunakan rumus sinus dan kosinus jumlah dua sudut, selisih dua sudut, dan sudut ganda untuk menghitung sinus dan kosinus sudut tertentu
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A. RUMUS JUMLAH DAN SELISIH DUA SUDUT
a
A(1, 0)
B
Misalkan kita punya lingkaran yang beradius 1 satuan
C
b
Koordinat:B(cos a, sin a) dan C(cos b, sin b)
Pythagoras: BC2 = x2 + y2
BC2 = (cos a – cos b)2 + (sin a – sin b)2
= cos2a – 2 cos a cos b + cos2b + sin2a – 2 sin a sin b + sin2b
= 2 – 2 (cos a cos b + sin a sin b) . . . . (1)
Lalu segitiga diputar ke kanan sehinggatitk C menempel di sumbu x.
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A. JUMLAH DAN SELISIH DUA SUDUT
a
A(1, 0)
B
Misalkan kita punya lingkaran yang beradius 1 satuan
C
b
Koordinat:B(cos a, sin a) dan C(cos b, sin b)
Pythagoras: BC2 = x2 + y2
BC2 = (cos a – cos b)2 + (sin a – sin b)2
= cos2a – 2 cos a cos b + cos2b + sin2a – 2 sin a sin b + sin2b
= 2 – 2 (cos a cos b + sin a sin b) . . . . (1)
Lalu segitiga diputar ke kanan sehinggatitk C menempel di sumbu x.
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A. JUMLAH DAN SELISIH DUA SUDUT
a|
C|(1, 0)
B|
A|
b
Koordinat:B|(cos (a–b), sin (a–b)) dan C|(1, 0)
Pythagoras: (B|C|)2 = x2 + y2
(B|C|)2 = (cos (a–b) – 1)2 + (sin (a–b) – 0)2
= cos2(a–b) – 2 cos (a–b) + 1 + sin2(a–b)
= cos2(a–b) + sin2(a–b) + 1 – 2 cos (a–b)
dari pers (1) dan (2) didapat:2 – 2 (cos a cos b + sin a sin b) = 2 – 2 cos (a–b)
= 2 – 2 cos (a–b) . . . . . . (2)
bababa sinsincoscos)(cos
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bababa sinsincoscos)(cos
Kalau cos (a + b) = ??
cos (a + b) = cos (a – (–b))
= cos a cos (–b) + sin a sin (–b)
= cos a cos b – sin a sin b
Untuk menghafalnya:
bababa sinsincoscos)(cos
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Contoh
Hitunglah cos 15o = ?
Jawab:
Ubah 15o ke sudut istimewa
15o = 45o – 30o atau 15o = 60o – 45o
cos 15o = cos (45o – 30o)
= cos 45o cos 30o + sin 45o sin 30o
2
1.
2
2
2
3.
2
2
4
26
bababa sinsincoscos)(cos
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Kerjakan
Classroom activities
Page 129
Nomor 1, 2, 3, 4
bababa sinsincoscos)(cos
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aaa oo sin2)45(cos)45(cos
Soal tambahan:
1. Hitunglah cos (15o – a) cos (15o + a) – sin (15o – a) sin (15o + a) = ?
2. cos (/2 + a) cos (/6 + a) + sin (/2 + a) sin (/6 + a) = ?
3. Buktikan
4. Diketahui dan ada di kuadran I , cos = 4/5 , dan cos = 24/25
Hitunglah cos ( + ) + 5 cos ( – ) = ?
5. Hitunglah cos 75o + cos 105o = ?
6. Buktikan cos (90 + a) = –sin a
7. Buktikan cos (180 + a) = –cos a
8. Buktikan cos (270 – a) = –sin a
9. Sederhanakan cos 25o cos 10o + sin 25o sin 10o
10. Hitunglah cos 140o cos 50o + sin 140o sin 50o = ?
11. Buktikan cos a – cos (a – 120o) – cos (a – 240o) = 2 cos a
12. Buktikan baba
batantan1
coscos
)(cos
bababa sinsincoscos)(cos
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Rumus: sin (a+b) & sin (a–b) page 129
di kuadran 1: sin x = cos (90o–x)
maka sin (a+b) = cos [90o – (a+b)]
= cos [(90o–a) – b]
= cos (90o–a) cos b + sin (90o–a) sin b
= sin a cos b + cos a sin b
Ganti (a–b) dengan (a+(–b)) didapat:
sin (a–b) = sin a cos b – cos a sin b
Untuk menghafalnya:
bsinacosbcosasinb)(asin
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Contoh
Hitunglah sin 173o = ?
Jawab:
173o = 120o + 53o
sin 173o = sin (120o + 53o)
= sin 120o cos 53o + cos 120o sin 53o
5
4
2
1
5
3
2
3..
10
433
bsinacosbcosasinb)(asin
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Kerjakan
Classroom activities
Page 130
Nomor 1 , 2 c , 3 b
bsinacosbcosasinb)(asin
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Soal tambahan:
1. Sederhanakan sin 3a cos 2a + cos 3a sin 2a
2. Sederhanakan sin 52o cos 14o – sin 14o cos 52o
3. Buktikan 214
3195sin o
4. Buktikan sin (a+b) . sin (a–b) = sin2a – sin2b
5. Buktikan sin2a – sin2b = cos2b – cos2a
6. Jika a, b lancip, cos a = 0,6 dan tan b = 2,4 hitung sin (a–b) = ?
bsinacosbcosasinb)(asin
7. Dari buku Mandiri hal. 40, kerjakan soal no: 6, 9, 10
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Rumus: tan (a+b) & tan (a–b) page 131
ingat: tan x = sin x / cos x
)(cos
)(sin)(tan
ba
baba
baba
baba
sinsincoscos
sincoscossin
Bagi dgn cos a cos b :
bababa
bababa
coscossinsincoscos
coscossincoscossin
ba
baba
tantan1
tantan)(tan
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Rumus: tan (a+b) & tan (a–b) page 131
Kalau tan (a–b) = ?
Untuk menghafalnya:
btanatan1
btanatanb)(atan
(a–b) = (a + (–b))
)(tantan1
)(tantan))((tan
ba
baba
ba
baba
tantan1
tantan)(tan
ba
ba
tantan1
tantan
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Contoh
Hitunglah tan 187o = ?
Jawab:
187o = 150o + 37o
tan 187o = tan (150o + 37o)
btanatan1
btanatanb)(atan
oo
oo
37tan150tan1
37tan150tan
43
.3
31
43
33
1233
1212
12349
)34(3
349
34
34
39
32548
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Kerjakan
Exercises
Page 133
Nomor genap saja
btanatan1
btanatanb)(atan