Ecuaciones trigonometricas
-
Upload
henry-villalba -
Category
Documents
-
view
230 -
download
12
Transcript of Ecuaciones trigonometricas
![Page 1: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/1.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
![Page 2: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/2.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
INTEGRANTES ‘’5to G’’
Brayan Choque Chura
Anthony Chambilla Chuquimia
Erick Gallegos Colana
Edwin Huamolle Mamani
Franklin Juárez Musaja
![Page 3: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/3.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
![Page 4: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/4.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
![Page 5: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/5.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
2Cos 3𝑥 + 3 = 0
Cos 3𝑥 = 𝑎𝑟𝑐 Cos −3
2
3𝑥 = 𝜋 − 𝑎𝑟𝑐 Cos3
2
3𝑥 = π −𝜋
6
∴ 𝒙 =𝟓𝝅
𝟏𝟖
![Page 6: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/6.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
2Sen𝑥 − 3 = 0
𝑥 = 𝑛𝜋 + −1 𝑛𝑎𝑟𝑐 Sen3
2
𝜋
3
∴ 𝐱 = 𝐧𝝅 + −𝟏 𝒏.𝝅
𝟑
![Page 7: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/7.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
![Page 8: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/8.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
Cotg 3𝑥 − 𝟏 = 𝟎 → Cotg 3𝑥 − 𝟏
𝑥 =𝑛𝜋 + 𝑎𝑟𝑐 Cotg 1
3
𝑥 =𝑛𝜋 +
𝜋4
3
𝑥 =𝑛𝜋
3+
𝜋
12 ∴ 𝒙 = 𝟔𝟎°𝒏 + 𝟏𝟓° ∴ 𝑪, 𝑺. = 𝟏𝟓°; 𝟕𝟓°; 𝟏𝟑𝟓°
![Page 9: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/9.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
𝟒𝐒𝐞𝐧𝒙𝐂𝐨𝐬𝒙 − 𝟏 = 𝟎
2.2 Sen 𝑥 Cos 𝑥 − 1 = 0
2 Sen2𝑥 − 1 = 0
Sen 2𝑥 =1
2
2𝑥 = 30°; 150°; 390°;…
𝑥 = 15°; 75°; 195°;…
∴ 𝒙 = 𝟏𝟓°
𝐒𝐞𝐧𝟐𝒙 = 𝟐𝐒𝐞𝐧𝒙𝐂𝐨𝐬 𝒙
![Page 10: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/10.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
𝒂𝒓𝒄𝐒𝐞𝐧 𝟐𝒙 + 𝟏 =𝝅
𝟔
𝒂𝒓𝒄 𝐒𝐞𝐧𝒙 = 𝒚 → 𝐒𝐞𝐧𝒚 = 𝒙
𝑎𝑟𝑐 Sen 2𝑥 + 1 =𝜋
6
Sen𝜋
6= 2𝑥 + 1
1
2= 2𝑥 + 1 → 𝒙 = −
𝟏
𝟐
![Page 11: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/11.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
![Page 12: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/12.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
T𝑔 2𝑥 = 1
2𝑥 = 𝑎𝑟𝑐 Tg 1
2𝑥 = 𝑎𝑟𝑐 Tg 1
2𝑥 =𝜋
4∴=
𝝅
𝟖SOLUCION
PRINCIPAL
![Page 13: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/13.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
Cos 2𝑥 =2
2
𝑥 =2𝑛𝜋 ± 𝑎𝑟𝑐 Cos
22
2
𝑥 =2𝑛𝜋 ±
𝜋4
2⇒ ∴ 𝒙 = 𝒏𝝅 ±
𝝅
𝟖SOLUCION
GENERAL
![Page 14: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/14.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
2 = Sec 2𝑥 → Sec 2𝑥 = 2
𝑥 =2𝑛𝜋 ± 𝑎𝑟𝑐 Cos 2
2
𝑥 =2𝑛𝜋 ±
𝜋3
2
𝑥 = 2𝑛 ±𝜋
6
∴ 𝒙 = 𝟏𝟖𝟎°𝒏 ± 𝟑𝟎°SOLUCION
GENERAL
𝑥 = 180° 0 − 30° = −30°
𝑥 = 180° 0 + 30° = 30°
𝑥 = 180° 1 − 30° = 150°
𝑥 = 180° 1 + 30° = 210°
𝑥 = 180° 2 − 30° = 330°
𝑥 = 180° 2 + 30° = 390°
NO!!
NO!!
∴ 𝑪. 𝑺 = 𝟑𝟎°; 𝟏𝟓𝟎°; 𝟐𝟏𝟎°; 𝟑𝟑𝟎°
![Page 15: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/15.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
𝐒𝐞𝐧𝟑𝒙 + 𝟑𝐂𝐨𝐬 𝟑𝒙 = 𝟐
Sen 𝛼 + 𝛽 = Sen𝛼 Cos 𝛽 + Cos 𝛼 + Sen𝛽
Sen3𝑥 + 3Cos 3𝑥 = 2
1
2Sen 3𝑥 +
3
2Cos 3𝑥 = 1
Sen3𝑥.1
2+ Cos 3𝑥.
3
2= 1
Sen3𝑥 . Cos 60° + Cos 3𝑥 . Sen 60° = 1
Sen 3𝑥 + 60° = Sen90°
3𝑥 + 60° = 90°
∴ 𝒙 = 𝟏𝟎° SOLUCION
PRINCIPAL
![Page 16: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/16.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
Sec 𝑎𝑟𝑐 Sen 2𝑥 = Sec 60°
𝑎𝑟𝑐 Sen 2𝑥 = 60°
Sen 60° = 2𝑥
3
2= 2𝑥
∴ 𝒙 =𝟑
𝟒
![Page 17: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/17.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
2Cos2 𝑥 − 5Cos 𝑥 + 2 = 0
Cos 𝑥
2Cos 𝑥
−2
−1
Cos 𝑥 − 2 2Cos 𝑥 − 1 = 0
Cos 𝑥 − 2 = 0
Cos 𝑥 = 2
2Cos 𝑥 − 1 = 0
Cos 𝑥 =1
2 ∴ 𝒙 = 𝟔𝟎°
![Page 18: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/18.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
𝑆𝑒𝑛2𝑥 + Cos 𝑥 = 0
2Sen 𝑥 Cos 𝑥 + Cos 𝑥 = 0
cos 𝑥 2Sen 𝑥 + 1 = 0
Cos 𝑥 = 0
𝒙 = 𝟗𝟎°
Sen 𝑥 + 1 = 0
Sen𝑥 = −1
2
𝑥 = 𝟐𝟏𝟎°
∴ 𝒙 = 𝟗𝟎°
![Page 19: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/19.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
𝑎𝑟𝑐 Sen 𝑥 + 1 = 𝛼 → Sen𝛼 = 𝑥 + 1
𝑎𝑟𝑐 Cos 1 − 𝑥 = 𝛽 → Cos𝛽 = 1 − 𝑥
𝛼 + 𝛽 =𝜋
2
Sen𝛼 = Cos 𝛽
𝑥 + 1 = 1 − 𝑥
2𝑥 = 0
∴ 𝒙 = 𝟎
![Page 20: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/20.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
4Cos 𝑥 − 31
Cos 𝑥= 2
Sen 𝑥
Cos 𝑥
4cos 𝑥 − 3Sec 𝑥 = 2T𝑔 𝑥
4Cos2 𝑥 − 3
Cos 𝑥=
2Sen 𝑥
Cos 𝑥
4Cos3 𝑥 − 3
Cos 3𝑥=
2Sen 𝑥 Cos 𝑥
Sen 2𝑥
3𝑥 + 2𝑥 = 90° ∴ 𝒙 = 𝟏𝟖°
![Page 21: Ecuaciones trigonometricas](https://reader033.fdocuments.co/reader033/viewer/2022052201/559413ec1a28ab8c2b8b45c1/html5/thumbnails/21.jpg)
425
13
0011 0010 1010 1101 0001 0100 1011
Tg2 𝑥 + 2Tg 𝑥 − 1 = 0
2Tg 𝑥 = 1 − T𝑔2 𝑥
2Tg 𝑥
1 − Tg2 𝑥= 1
Tg 2𝑥 = 1
𝑥 =𝑛𝜋 + 𝑎𝑟𝑐 Tg 1
2
𝑥 =𝑛𝜋 +
𝜋4
2→ 𝑥 =
𝑛𝜋
2+
𝜋
8
∴ 𝒙 =𝟒𝒏 + 𝟏
𝟖𝝅