TRIGONOMETRI KELAS 10

1. MATEMATIKA
TRIGONOMETRI
Disusun oleh :
NURFAIZAH
MATERI PELAJARAN
TENTANG

2. TRIGONOMETRI
Konsep Umum : PHYTAGORAS
𝑑𝑒𝑝𝑎𝑛
Sin 𝖺 = =
3
𝑚𝑖𝑟𝑖𝑛𝑔 5
𝑚𝑖𝑟𝑖𝑛
𝑔
=
Cos 𝖺 = 𝑠𝑎𝑚𝑝𝑖𝑛𝑔 4
5
Tan 𝖺 =
𝑑𝑒𝑝𝑎𝑛
= 3
𝑠𝑎𝑚𝑝𝑖𝑛𝑔 4
Sec 𝖺 =
𝐶𝑜𝑠 𝖺 4
5
1
= 1
= 5
4
Cosec 𝖺 =
𝑆𝑖𝑛 𝖺 3
5
1
= 1
= 5
3
Cot 𝖺 =
𝑇𝑎𝑛 𝖺 3
4
1
= 1
= 4
3

3. KUNCI SUKSES MATERI
TRIGONOMETRI
WAJIB
DI HAFALKAN !!!

4. NILAI PERBANDINGAN
TRIGONOMETRI
RUMUS IDENTITAS
TRIGONOMETRI
RUMUS 1
PENJUMLAHAN
SUDUT (sinus/cosinus)
RUMUS 2
JUMLAH & SELISIH
SUDUT (sinus/cosinus)
RUMUS 3
PERKALIAN SUDUT
(sinus/cosinus)
RUMUS 4
TRIGONOMETRI

5. NILAI PERBANDINGAN
TRIGONOMETRI
Kuadran II
(180°− 𝖺)
90° < 𝐱 < 180°
Sin +
Kuadran I
(90°− 𝖺)
< 90°
All +
Kuadran III
(180°+ 𝖺)
180° < 𝐱 < 360°
Tan +
Kuadran IV
(360°− 𝖺)
Cos +

6. Contoh Soal
A.Sin 140° = … B.Cos 240° = …
Penyelesaian:
A. Sin 135° =
Kuadran II Sin
(180°− 𝖺) = 135° Sin
(180° − 𝟒𝟓°) = 135°
𝖺 = 𝟒𝟎°
Penyelesaian:
B. Cos 240° =
Kuadran III Cos (18𝟎
°+ 𝖺) = 240° Cos (1𝟖𝟎
° + 𝟔𝟎°) = 240°
𝖺 = 6𝟎°
Sin 𝟒𝟓° = Cos 60° =

7. RUMUS IDENTITAS
TRIGONOMETRI
1. 𝑺𝒊𝒏𝟐 𝑨 + 𝑪𝒐𝒔𝟐 𝑨 = 𝟏
2. 1+𝑻𝒂𝒏𝟐𝑨 =
𝟏
𝑪𝒐𝒔𝟐𝑨
= 𝑺𝒆𝒄𝟐 𝑨
3. 1+𝑪𝒐𝒕𝟐𝑨 =
𝟏
𝑺𝒆𝒄𝟐𝑨
= 𝑪𝒐𝒔𝒆𝒄𝟐 𝑨
Tidak ada contoh soal

8. RUMUS 1
PENJUMLAHAN SUDUT
(SINUS/COSINUS)
SINUS
1. 𝐒𝐢𝐧 𝐀 + 𝐁 = 𝐒𝐢𝐧𝐀. 𝐂𝐨𝐬𝐁 + 𝐂𝐨𝐬𝐀. 𝐒𝐢𝐧𝐁
2. 𝐒𝐢𝐧 𝐀 − 𝐁 = 𝐒𝐢𝐧𝐀. 𝐂𝐨𝐬𝐁 − 𝐂𝐨𝐬𝐀. 𝐒𝐢𝐧𝐁
COSINUS
1. Cos 𝐀 + 𝐁 = 𝐂𝐨𝐬𝐀. 𝐂𝐨𝐬𝐁 − 𝐒𝐢𝐧𝐀. 𝐒𝐢𝐧𝐁
2. Cos 𝐀 − 𝐁 = 𝐂𝐨𝐬𝐀. 𝐂𝐨𝐬𝐁 + 𝐒𝐢𝐧𝐀. 𝐒𝐢𝐧𝐁

9. = 1/2
= 1/4
2. 1/2+1/2 2.1/2 3
2+1/4 6
B. Sin 36°. Cos 24° + Cos 36°. Sin 24°
= Sin (36° + 24°)
= Sin 60°
= 1/2 3
Contoh Soal
Tentukan Nilai dari:
A.Sin (45° + 𝟔𝟎°) !
B.Sin 36°. 𝐂𝐨𝐬 𝟐𝟒° + 𝐂𝐨𝐬 𝟑𝟔°. 𝐒𝐢𝐧 𝟐𝟒° !
Penyelesaian:
A. Sin (45° + 60°)
= Sin 45°.Cos 60° + Cos 45°. Sin 60°

10. RUMUS 2
JUMLAH & SELISIH SUDUT
(SINUS/COSINUS)
𝐀 − 𝐁
𝐀 − 𝐁
SINUS
1. 𝐒𝐢𝐧𝐀 + 𝐒𝐢𝐧𝐁 = 𝟐. 𝐒𝐢𝐧 𝟏/𝟐 𝐀 + 𝐁 . 𝐂𝐨𝐬 𝟏/𝟐
2. 𝐒𝐢𝐧𝐀 − 𝐒𝐢𝐧𝐁 = −𝟐. 𝐒𝐢𝐧 𝟏/𝟐 𝐀 + 𝐁 . 𝐂𝐨𝐬 𝟏/𝟐
COSINUS
1. Cos𝐀 + 𝐂𝐨𝐬𝐁 = 𝟐. 𝐂𝐨𝐬 𝟏/𝟐 𝐀 + 𝐁 . 𝐂𝐨𝐬 𝟏/𝟐
2. 𝐂𝐨𝐬𝐀 − 𝐂𝐨𝐬𝐁 = −𝟐. 𝐒𝐢𝐧 𝟏/𝟐 𝐀 + 𝐁 . 𝐒𝐢𝐧 𝟏/𝟐
𝐀 − 𝐁
𝐀 − 𝐁

11. Penyelesaian:
Cos 90° + 𝐶𝑜𝑠 30°
= 2. 𝐶𝑜𝑠 1/2 𝐴 + 𝐵 . 𝐶𝑜𝑠 1/2 𝐴 − 𝐵
= 2. 𝐶𝑜𝑠 1/2 90° + 30° . 𝐶𝑜𝑠 1/2 90° − 30°
= 2. 𝐶𝑜𝑠 1/2 . 120° . 𝐶𝑜𝑠 1/2 . 60°
= 2. 𝐶𝑜𝑠 60° . 𝐶𝑜𝑠 30°
= 2. 1/2 . 1/2 3
= 1 . 1/2 3
Contoh Soal
Tentukan Nilai dari :
Cos 90° + 𝐂𝐨𝐬 𝟑𝟎° !

12. RUMUS 3
PERKALIAN SUDUT
(SINUS/COSINUS)
1. 𝟐. 𝐒𝐢𝐧𝐀. 𝐂𝐨𝐬𝐁 = 𝐒𝐢𝐧 𝐀 + 𝐁
2. 𝟐. 𝐂𝐨𝐬𝐀. 𝐒𝐢𝐧𝐁 = 𝐒𝐢𝐧 𝐀 + 𝐁
3. 2.Cos𝐀. 𝐂𝐨𝐬𝐁 = 𝐂𝐨𝐬 𝐀 + 𝐁
4. −𝟐. 𝐂𝐨𝐬𝐀. 𝐂𝐨𝐬𝐁 = 𝐂𝐨𝐬 𝐀 + 𝐁
+ 𝐒𝐢𝐧 𝐀 − 𝐁
− 𝐒𝐢𝐧 𝐀 − 𝐁
+ 𝐂𝐨𝐬 𝐀 − 𝐁
− 𝐒𝐢𝐧 𝐀 − 𝐁

13. Penyelesaian:
2. Cos 60°. Sin 60°
= 2. Sin 60° + 30° − Sin 60° − 30°
= 2. Sin 90° − Sin 30°
= 2. 1 - 1/2
= 2 - 1/2
Contoh Soal
𝐓𝐞𝐧𝐭𝐮𝐤𝐚𝐧 𝐍𝐢𝐥𝐚𝐢 𝐝𝐚𝐫𝐢 ∶
𝟐. 𝐂𝐨𝐬 𝟔𝟎°. 𝐒𝐢𝐧 𝟑𝟎°

14. RUMUS 4
TRIGONOMETRI
1. 𝑺𝒊𝒏𝟐
2. 𝑪𝒐𝒔𝟐
3. 𝑻𝒂𝒏𝟐 𝖺 =
𝖺 = 𝟐. 𝑺𝒊𝒏 𝖺 . 𝑪𝒐𝒔
𝖺
𝖺 = 𝑪𝒐𝒔𝟐 𝖺 −𝑺𝒊𝒏𝟐 𝖺
=2.𝑪𝒐𝒔𝟐 𝖺 −1
𝟐.𝑻𝒂𝒏 𝖺
𝟏−𝑻𝒂𝒏𝟐𝖺

15. Contoh Soal
Penyelesaian:
𝑆𝑖𝑛230° =
=
=
𝑆𝑖𝑛 30° 2
1/2
2
1/4
Bentuk sederhana dari :
𝑺𝒊𝒏𝟐𝟑𝟎° !

16. LATIHAN
SOAL
1. Cos (270°)
2. Sin (204°)
3. Tan (60°)
4. Sec (167°)
5. Sin260°+Cos260°
6. 2. 𝐶𝑜𝑠260°+2. 𝑆𝑖𝑛245+3. 𝑇𝑎𝑛245°
7. Sin 135°
8. Cos(90°-30°)
9. Cos 60°-Cos 30°
10.Jika f(x) = (Cos x + Sin x) . Tentukan nilai f(x),
jika x = 45°

17. PEMBAHASAN
SOAL
1. Cos (270°) = KIII
Cos (180°+𝖺) = 270°
Cos (180°+𝟗𝟎°) = 270°
𝖺 = 𝟗𝟎°
Cos 𝟗𝟎° = 𝟎
2. Sin (204°) = KIII
Sin (180°+𝖺) = 204°
Sin (180°+𝟐𝟒°) = 204°
𝖺 = 𝟐𝟒° (−)
3. Tan (60°) = KI
Tan (90°− 𝖺) = 60°
Tan (90° − 𝟑𝟎°) = 60°
𝖺 = 𝟑𝟎°
Tan 3𝟎° = 𝟏/𝟑 𝟑
4. Sec (167°) = KII
Sec (180° −𝖺) = 167°
Sin (180° − 𝟏𝟑°) = 167°
𝖺 = 𝟏𝟑° (+)

18. 5. 𝐒𝐢𝐧𝟐60°+𝐂𝐨𝐬𝟐60°
= (Sin 60°)𝟐+(Cos 60°)𝟐
= ( 𝟏/𝟐 𝟑)𝟐+( 𝟏/𝟐)𝟐
= 𝟏/𝟒 . 𝟑 + 𝟏/𝟒
= 𝟑/𝟒 + 𝟏/𝟒
= 𝟒/𝟒
= 1
6. 2. 𝐂𝐨𝐬𝟐60°+2. 𝐒𝐢𝐧𝟐45+3. 𝐓𝐚𝐧𝟐45°
= 2(Cos 60°)𝟐 + 2(Sin 60°)𝟐+3(Tan 𝟒𝟓°)𝟐
= 2( 𝟏/𝟐)𝟐 + 𝟐( 𝟏/𝟐 𝟐)𝟐+𝟑(𝟏)𝟐
= 2(𝟏/𝟒) +2(𝟏/𝟒.2)+3.1
= 𝟐/𝟒+𝟐/𝟒.4+3
= 𝟐/𝟒+2+3
= 𝟐/𝟒 + 𝟓

19. 7. Sin135° =(Sin90+45)
=𝐒𝐢𝐧 𝐀 + 𝐁 = 𝐒𝐢𝐧𝐀. 𝐂𝐨𝐬𝐁 + 𝐂𝐨𝐬𝐀. 𝐒𝐢𝐧𝐁
=Sin 90°.Cos 45°+Cos 90°.Sin 45°
=1.𝟏/𝟐 𝟐+0.𝟏/𝟐 𝟐
=𝟏/𝟐 𝟐
8. Cos(90°-30°)
=Cos 𝐀 − 𝐁 = 𝐂𝐨𝐬𝐀. 𝐂𝐨𝐬𝐁 + 𝐒𝐢𝐧𝐀. 𝐒𝐢𝐧𝐁
=Cos90°.Cos 45°+Sin90°.Sin 45°
=𝟏/𝟐
=𝟏/𝟐
𝟐.0+𝟏/𝟐 𝟑. 𝟏
𝟑

20. 9. Cos60°-Cos30°
=𝐂𝐨𝐬𝐀 − 𝐂𝐨𝐬𝐁 = −𝟐. 𝐒𝐢𝐧 𝟏/𝟐 𝐀 + 𝐁 . 𝐒𝐢𝐧 𝟏/𝟐 𝐀 − 𝐁
=−𝟐. 𝐒𝐢𝐧 𝟏/𝟐 𝟔𝟎° + 𝟑𝟎° . 𝐒𝐢𝐧 𝟏/𝟐 𝟔𝟎° − 𝟑𝟎°
=−𝟐. 𝐒𝐢𝐧 𝟏/𝟐 𝟗𝟎° . 𝐒𝐢𝐧 𝟏/𝟐 𝟑𝟎°
=−𝟐. 𝐒𝐢𝐧 𝟒𝟓°. 𝐒𝐢𝐧 𝟑𝟎°
=−𝟐. 𝟏/𝟐 𝟐. 𝐒𝐢𝐧 𝟏𝟓°
=− 𝟐/𝟐 . −𝟐. 𝐒𝐢𝐧 𝟏𝟓°
=−𝟏. −𝟐. 𝐒𝐢𝐧 𝟏𝟓°
=𝟐. 𝐒𝐢𝐧 𝟏𝟓°

21. 10.Jikaf(x) =(Cosx+Sinx) .Tentukannilai f(x),
jikax=45°
F (x) =(Cosx +Sinx) ( Cosx+Sinx)
=𝑪𝒐𝒔𝟐𝒙 + 𝑺𝒊𝒏 𝒙. 𝐂𝐨𝐬 𝒙+ 𝑺𝒊𝒏 𝒙. 𝐂𝐨𝐬 𝒙+ 𝑺𝒊𝒏𝟐𝒙
=𝑪𝒐𝒔𝟐𝒙 + 𝑺𝒊𝒏𝟐𝒙 + 𝑺𝒊𝒏 𝒙. 𝐂𝐨𝐬 𝒙+ 𝑺𝒊𝒏 𝒙. 𝐂𝐨𝐬 𝒙
=1+𝑺𝒊𝒏 𝒙. 𝐂𝐨𝐬 𝒙+ 𝑺𝒊𝒏 𝒙. 𝐂𝐨𝐬 𝒙
=1+(Sin (A+B)+Sin(A-B))
=1+(Sin (45°+45°)+Sin(45°-45°))
=1+(Sin 90° +Sin90°)
=1+1+0
=2