Sunday, December 6, 2020

Soal Matematika Wajib SMA - Trigonometri

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Berikut Link-link Soal-soal SMA Matematika Wajib

 

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TRIGONOMETRI

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Ukuran Sudut (Derajat dan Radian)

 

 

Satu radian diartikan sebagai besar ukuran sudut pusat α yang panjang busurnya sama dengan jari-jari, perhatikan Gambar. Jika AOB = α dan AB = OA = OB, maka

 

α =  = 1 radian.

 

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360o = 2π rad atau 1o =  rad

atau 1 rad =   57,3o

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Sudut istimewa yang sering digunakan

 

 

 

Perhatikan hubungan secara aljabar antara derajat dengan radian berikut ini.

 

 putaran = 1 ×360o = 90o

 

atau

 

90o = 90 ×  rad =  rad

 

Latihan

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Tentukan nilai kebenaran (benar atau salah) setiap pernyataan di bawah ini. Berikan penjelasan untuk setiap jawaban yang diberikan.

a.       putaran = 0,33π rad = 60o

b.      150o = putaran =  rad

c.      4 π rad = 792o = 2,4 putaran

d.      1.500o = 8π rad = 4 putaran

e.      Seorang atlet berlari mengelilingi lintasan A berbentuk lingkaran sebanyak 2 putaran. Hal itu sama saja dengan atlet berlari mengelilingi satu kali lintasan B berbentuk lingkaran yang jari-jarinya 2 kali jarijari lintasan A.

 

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Contoh

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Gambarkan sudut-sudut baku di bawah ini, dan tentukan posisi setiap sudut pada koordinat kartesius.

a. 60o

b. –45o

 

Jawab

 

 

 

 

Latihan

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Gambarkan setiap ukuran sudut di bawah ini dalam koordinat kartesius.

a. 120o

b. 600o

c. 270o

d. –240o

e. 330o

f. –800o

 

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Latihan

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1.    Diketahui besar sudut α kurang dari 90o dan besar sudut θ lebih dari atau sama dengan 90o dan kurang dari 180o. Analisislah kebenaran setiap pernyataan berikut ini.

         a.   2 α ≥ 90o

         b.   θ α ≥ 30o

         c.   2a +  θ ≥ 90o

         d.   Ada nilai α dan θ yang memenuhi persamaan 2θ – 2 α = θ + α

 

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2.      Berikut ini merupakan besar sudut dalam satuan derajat, tentukan kuadran setiap sudut tersebut.

         a. 90o                          d. 800o

         b. 135o                        e. –270o

         c. 225o                        f. 1.800o

        Selanjutnya, nyatakan setiap sudut di atas dalam satuan radian.

 

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3.    Tentukan (dalam satuan derajat dan radian) untuk setiap rotasi berikut.

        a.  putaran

 

        b.  putaran

 

        c.  putaran

       

        d.  putaran

       

        e.  putaran

       

        f.  putaran

 

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4.    Nyatakan dalam radian besar sudut yang dibentuk untuk setiap penunjukan waktu berikut.

a. 12.05                        d. 05.57

b. 00.15                       e. 20.27

        c. 16.53                                       f. 07.30

 

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5.    Misalkan θ merupakan sudut lancip dan sudut β adalah sudut tumpul.

Perhatikan kombinasi setiap sudut dan kedua sudut tersebut dan tentukan kuadrannya.

a. 3θ              c. θ + β

b. 2 β             d. 2 β θ

 

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6.    Perhatikan pergerakan jarum jam. Berapa kali (jika ada) dalam 1 hari terbentuk sudut-sudut di bawah ini?

        a. 90o               c. 30o

        b. 180o                         d. 120o

 

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8.    Ubahlah sudut-sudut berikut ke bentuk derajat

 

        a.  rad

 

        b.  rad

 

        c.  rad

 

        d.  rad

 

        e.  rad

 

        f.  rad

 

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TRIGONOMETRI

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Perbandingan Trigonometri pada Segitiga Siku-Siku

 

 

Jika diketahui segitiga ABC siku-siku maka diidentifikasikan perbandingan trigonometri sebagai berikut :

 

 

Contoh

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Diberikan segitiga siku-siku ABC, sin A = . Tentukan cos A, tan A, sin C, cos C, dan cot C.

 

Jawab

 

Kita memerlukan panjang sisi AB. Dengan menggunakan Teorema Pythagoras, diperoleh

 

AB2 = AC2 – BC2

AB =

AB =

AB =

AB =

 

Jadi, kita memperoleh panjang sisi AB =

maka

 

cos A =

 

tan A =

 

Sin C =

 

cos C =

 

cot C =

 

 

Perlu Diingat

Panjang sisi miring adalah sisi terpanjang pada suatu segitiga siku-siku. Akibatnya nilai sinus dan cosinus selalu kurang dari 1 (pada kondisi khusus akan bernilai 1).

 

Latihan

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1.    Tentukan nilai sinus, cosinus, dan tangen untuk s            udut P dan R pada setiap segitiga siku-siku di bawah ini. Nyatakan jawaban kamu dalam bentuk paling sederhana.

       

       

       

 

 

 

 

2.      Pada suatu segitiga siku-siku ABC, dengan

         B = 90o, AB = 24 cm, dan BC = 7 cm, hitung:

         a. sin A dan cos A

         b. sin C, cos C, dan tan C

 

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Contoh

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Dua orang guru dengan tinggi badan yang sama yaitu 170 cm sedang berdiri memandang puncak tiang bendera di sekolahnya. Guru pertama berdiri tepat 10 m di depan guru kedua.

Jika sudut elevasi guru pertama 60o dan guru kedua 30o dapatkah kamu menghitung tinggi tiang bendera tersebut?

 

Jawab

Dimana:

AC = tinggi tiang bendera

DG = tinggi guru pertama

EF = tinggi guru kedua

DE = jarak kedua guru

 

tan 60o = 

BG =   

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tan 30o =  

tan 30o =

AB = (10 + BG) × tan 30o

AB =

AB × tan 60o = (10 × tan 60o + AB) × tan 30o

AB × tan 60o = 10 × tan 60o × tan 30o + AB × tan 30o

AB × tan 60o AB × tan 30o = 10 × tan 60o × tan 30o

AB × (tan 60o tan 30o) = 10 × tan 60o × tan 30o

AB =

 

Jadi, tinggi tiang bendera adalah

 

AC = AB + BC atau AC =  m

 

Latihan

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1.    Untuk setiap nilai perbandingan trigonometri yang diberikan di bawah ini, dengan setiap sudut merupakan sudut lancip, tentukan nilai 5 macam perbandingan trigonometri lainnya.

 

        a. sin A =

 

        b. 15 × cot A = 8

 

        c. sec θ =

 

        d. tan α =

 

        e. sin α =

 

        f. tan α =

 

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2.    Pada sebuah segitiga KLM, dengan siku-siku di L, jika sin M =  dan panjang sisi KL =  cm, tentukan panjang sisi segitiga yang lain dan nilai perbandingan trigonometri  lainnya.

 

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3.    Jika cot θ = , hitung nilai dari :

 

        a.   

 

        b.   

 

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7.    Perhatikan segitiga siku-siku di bawah ini.

       

Tunjukkan bahwa

 

a)    (sin A)2 + (cos A)2 = 1

 

        b)    tan B =

 

        c)    (scs A)2 – (cot A)2 = 1

 

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8.      Dalam segitiga ABC, siku-siku di A diketahui panjang BC = a, (a adalah bilangan positif) dan

         cos ABC = Tentukan panjang garis tinggi AD.

 

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9.      Diketahui sin x + cos x = 1 dan tan x = 1, tentukan nilai sin x dan cos x.

 

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10.    Pada segitiga PQR, siku-siku di Q, PR + QR = 25 cm, dan PQ = 5 cm. Hitung nilai sin P, cos P, dan tan P.

 

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11.    Diketahui segitiga PRS, seperti gambar di samping ini. Panjang PQ =1, RQS = α rad dan RPS = β rad. Tentukan panjang sisi RS.

 

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TRIGONOMETRI

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Nilai perbandingan trigonometri untuk sudut-sudut istimewa

 

 

0o

30o

45o

60o

90o

-

 

Contoh

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Diberikan suatu segitiga siku-siku KLM, siku-siku di L. Jika LM = 5 cm, dan

M = 30o. Hitung:

a.      panjang KL dan MK,

b.      cos K,

 

Jawab

 

a.     cos 30o =

 

        cos 30o =

 

       

 

 

        MK =  cm

 

 

        tan 30o =

 

        tan 30o =

 

       

 

 

        MK =  cm

 

b.    karena L = 90o dan M = 30o, maka K = 60o. Akibatnya cos 60o =

 

Latihan

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1.    Diketahui segitiga RST, dengan S = 90o, T = 60o, dan ST = 6 cm.

        Hitung :

        a. Keliling segitiga RST

        b. (sin T)2 + (sin R)2

 

2.    Hitung nilai dari setiap pernyataan trigonometri berikut.

        a.     sin 60o × cos 30o + cos 60o × sin 30o

 

        b.    2(tan 45o)2 + (cos 30o) – (sin 60o)2

 

        c.    

 

        d.   

       

        e.   

 

 

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Contoh

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Diketahui sin (A B) = , cos (A + B) = ,

0o < (A + B) < 90o, A > B

Hitung sin A dan tan B.

 

Jawab :

sin (A B) =

A – B = 30

 

cos (A + B) =

A + B = 60

 

maka kita eliminasi

A – B = 30

A + B = 60

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2A = 90

A = 45

 

A – B = 30

45 – B = 30

B = 15

 

Latihan

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Diketahui sin (A B) = , cos (A + B) = ,

0o < (A + B) < 90o, A > B

Tentukan A dan B.

 

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Latihan

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1.    Manakah pernyataan yang bernilai benar untuk setiap pernyataan di bawah ini.

         a.   sin (A + B) = sin A + sin B

         b.   Nilai sin θ akan bergerak naik pada saat nilai θ juga menaik, untuk 0o ≤ θ ≤ 90o

         c.   Nilai cos θ akan bergerak naik pada saat nilai θ menurun, untuk 0o ≤ θ ≤ 90o

         d.   sin θ = cos θ, untuk setiap nilai θ

         e.   Nilai cot θ tidak terdefinisi pada saat θ = 0o

 

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2.    Jika , 0o < β < 90o hitung nilai β.

 

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3.    Jika sin x = a dan cos y = b dengan 0 < x < , dan  < y < π , maka hitung tan x + tan y.

 

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4.    Pada suatu segitiga ABC, diketahui a + b = 10,

        A = 30o, dan B = 45o. Tentukan panjang sisi b.

        (Petunjuk: Misalkan panjang sisi di depan A = a,

        di depan B = b, dan B = c).

 

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5.    Diketahui segitiga ABC, siku-siku di B, cos a = , dan tan  = 1, seperti gambar berikut.

       

Jika AD = a, hitung:

a. AC

        b. DC

 

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6.    Perhatikan gambar di bawah ini.

Buktikan

a. OC = sec θ

b. CD = tan θ

c. OE = csc θ

        d. DE = cot θ

 

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TRIGONOMETRI

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Jika 0o a 90o, maka berlaku.

a.   sin (90oa) = cos a

b.   cos (90oa) = sin a

c.   tan (90oa) = cot a

d.   csc (90oa) = sec a

e.   sec (90oa) = csc a

f.    cot (90oa) = tan a

 

Contoh

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Sederhanakan bentuk

 

Jawab

diketahui bahwa cot A = tan (90oA).

Akibatnya, cot 25o = tan (90o – 25o) = tan 65o.

Jadi ,

 

 

 

Contoh

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sin 3A = cos (A – 26o), dengan 3A adalah sudut lancip. Hitung A.

 

Jawab :

Diketahui sin 3A = cos (A – 26o). Karena 3A adalah

sudut lancip, maka sin 3A = cos (90o – 3A)

Akibatnya,

cos (90o – 3A) = cos (A – 26o)

(90o – 3A) = (A – 26o)

A = 29o

 

 

Latihan

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sin 5A = cos (A 40o), dengan 3A adalah sudut lancip. Hitung A.

 

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Perbandingan Trigonometri di Berbagai Kuadran

Sudut negatif :

 

 

 

 

 

 

 


Grafik Sinus

 

Latihan

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Tentukan

a.      sin 45o

b.      sin 135 o

c.       sin 225 o

d.      sin 315 o

e.      sin 30 o

f.       sin 120 o

g.      sin 210 o

h.      sin 300 o

i.        sin 60 o

j.        sin 150 o

 

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Grafik Cosinus

 

Latihan

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Tentukan

a.      cos 45o

b.      cos 135 o

c.       cos 225 o

d.      cos 315 o

e.      cos 30 o

f.       cos 120 o

g.      cos 210 o

h.      cos 300 o

i.        cos 60 o

j.        cos 150 o

 

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Grafik Tangen

 

 

Latihan

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Tentukan

a.      tan 45o

b.      tan 135 o

c.       tan 225 o

d.      tan 315 o

e.      tan 30 o

f.       tan 120 o

g.      tan 210 o

h.      tan 300 o

i.        tan 60 o

j.        tan 150 o

 

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Identitas Trigonometri

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Rumus-rumus identitas trigonometri adalah sebagai berikut :

 

 

 

Contoh

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Misalkan 0o < β < 90o dan tan β = 3

Hitung nilai sin  β dan cos β.

 

Jawab

sin  β

--------------------------

Dengan menggunakan definisi perbandingan dan identitas trigonometri, diperoleh cot β =

 

Akibatnya, 1 + cot2 α = csc2 α

 

1 +