trigonometri sma kelas xi ipa sem 1
TRANSCRIPT
http://meetabied.wordpress.com
Setelah menyaksikan tayangan ini anda dapat
Menyelesaikansoal-soal yang berkaitan
dengan jumlah dan selisih sudutserta sudut rangkap
http://meetabied.wordpress.com
Rumusjumlah dan selisih dua sudut
sin(α + β)
= sinα.cosβ + cosα.sinβ sin(α - β)
= sinα.cosβ - cosα.sinβ
http://meetabied.wordpress.com
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)
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
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 −=−
http://meetabied.wordpress.com
Rumus
jumlah dan selisih dua sudut cos(α + β)
= cosαcosβ - sinαsinβ
cos(α - β)
= cosαcosβ + sinαsinβ
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
2. a. –sina.sinb b. cosa.cosb c. sina.sinb d. 1 – tana.tanb e. 1 + tana.tanb
....bcos.acos
)bacos( =+
http://meetabied.wordpress.com
=
= 1 – tana.tanb → jawab d
=−bcos.acos
)bacos(
bcos.acos
bsin.asinbcos.acos +
bcos.acos
bsin.asin
bcos.acos
bcos.acos +
http://meetabied.wordpress.com
3. Tentukan
nilai cos56° + sin56°.tan28°
Bahasan: cos56° + sin56°.tan28°
= cos56° + sin56°.
= cos56° +
0
0
28cos
28sin
0
00
28cos
28sin.56sin
http://meetabied.wordpress.com
= 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
http://meetabied.wordpress.com
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°
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
Rumusjumlah dan selisih dua sudut
tan(α + β) =
tan(α - β) =
βαβα
tan.tan1
tantan
−+
βαβα
tan.tan1
tantan
+−
http://meetabied.wordpress.com
1. tan 105° = …. Bahasan:
tan105° = tan(60° + 45°)
oo
oo
45tan.60tan1
45tan60tan
−+=
1.31
13
−+=
x31
31
−+=
31
31
++
http://meetabied.wordpress.com
tan 105° = x
=
=
=
= -2 - √3
31
31
−+
31
31
++
31
)31( 2
−+
31
3321
−++
2
324
−+
http://meetabied.wordpress.com
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
−+
http://meetabied.wordpress.com
= -1 tan A + ½= -1 + ½tan A tan A - ½tan A = -1 - ½ ½tan p = -1½ Jadi, tan p = -3
21
21
.Atan1
Atan
−+
http://meetabied.wordpress.com
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
+−
http://meetabied.wordpress.com
= 1 tan p - ½ = 1 + ½tan p tan p - ½tan p = 1 + ½ ½tan p = 1½ Jadi, tan p = 3
21
21
.ptan1
ptan
+−
http://meetabied.wordpress.com
Rumus Sudut Rangkap
sin2a = 2 sina.cosa
contoh: 1. sin10° = 2sin5°.cos5°
2. sin6P = 2sin3P.cos3P
3. sin t = 2sin½t.cos½t
http://meetabied.wordpress.com
1.Diketahui cosα =
Nilai sin 2α =….
Bahasan:
cos α =
sinα =
α4
3
5
3
55
3
5
4
http://meetabied.wordpress.com
cos α =
sinα =
Jadi sin2α = 2sinα.cosα = 2. x =
α4
3
55
3
5
4
5
4
5
3
25
24
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
3. Jika sinx – cosx = p maka harga sin 2x =….
Bahasan: sinx – cosx = p
(sinx – cosx)2 = p2
sin2x – 2sinx.cosx + cos2x = p2
http://meetabied.wordpress.com
sin2x – 2sinx.cosx + cos2x = p2
sin2x + cos2x – 2sinx.cosx = p2
1 – sin2x = p2
1 – p2 = sin2x
Jadi, harga sin2x = 1 – p2
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
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 −
http://meetabied.wordpress.com
Rumus Sudut Rangkap
cos 2a = cos2a – sin2a
= 2cos2a – 1
= 1 – 2sin2a
http://meetabied.wordpress.com
1. Diketahui cosα =
maka cos 2α =….
Bahasan:
cos2α = 2cos2α - 1
= 2( )2 – 1
= - 1
= -
3
1
3
1
9
2
9
7
http://meetabied.wordpress.com
2. Diketahui sinx = ½ maka cos 2x =….
Bahasan:
cos2x = 1 – 2sin2x
= 1 – 2(½)2
= 1 – ½ = ½
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
• 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
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
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 −+
http://meetabied.wordpress.com
6. Buktikan:
Bahasan:
atanasin
acos121=−
=−asin
acos1aa
a
21
21
212
cossin2
)sin21(1 −−
acosasin2
asin2
21
21
212
=
http://meetabied.wordpress.com
=−asin
acos1
acosasin2
asin2
21
21
212
acos
asin
21
21
=
atan 21=
Terbukti : =−asin
acos1atan 2
1
http://meetabied.wordpress.com
Rumus Sudut Rangkap
tan 2a =
Contoh: 1. tan 20° =
2. tan 10x =
atan1
atan22−
02
0
10tan1
10tan2
−
x5tan1
x5tan22−
http://meetabied.wordpress.com
1. Jika tan A = 3 maka tan 2A =….
Bahasan:
tan 2A =
=
= =
Atan1
Atan.22−
231
3.2
−
8
6
− 4
3−
http://meetabied.wordpress.com
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
http://meetabied.wordpress.com
tan 2x =
=
=
Jadi, tan 2x =
25144
524
1 −
25144 25
524
−
119
5.24
− 119
120−
http://meetabied.wordpress.com
SELAMAT BELAJAR
http://meetabied.wordpress.com