6 + 2 sen x = 1

1. Clase 75 6 + 2 sen x = 1 Ejercicios sobre Ecuaciones trigonométricas 2 cos2x – cos x = 0 2 cos2x + 5 sen x = –1

2. cosx sen x cos x Revisión del estudio individual C) sen 2x = tan x 2 senxcosx = 2 senxcos2x = senx 2 senx (1 – sen2x) – sen x = 0 2 senx– 2 sen3x – sen x = 0 2 sen3x – sen x = 0 senx(2sen2x– 1) = 0

3. 1 2 2 senx =  2   x = kó x = + k 4 2 senx(2sen2x– 1) = 0 sen2x = sen x = 0 ó kZ

4. Ejercicio Resuelve las siguientes ecuaciones: a) 2 sen 2x cot x – cos 2x = 3 cos x sen 2x cos x b) = 2 sen2x sen x c) sen 2x = 2 – 2 cos 2x

5. cosx senx 1 5 2   = 3 3 3 a) 2 sen 2x cot x – cos 2x = 3 cos x 4senx cosx –(2cos2x – 1) = 3 cos x 4 cos2x – 2 cos2x + 1 = 3 cos x 2 cos2x – 3 cos x + 1 = 0 (2 cos x – 1)(cos x – 1) = 0 ó cos x = 1 cos x = x1= + 2k ; k x3= 2k x2= 2 – + 2k ; k

6. sen 2x cos x : 2 3 3   4 2 4 2 x1 = x2 = + k + k b) = 2 sen2x sen x 2 senx cosx cosx = 2 sen2x sen x 2 cos2 x = 2 sen2x cos2x – sen2x = 0 cos 2x = 0 k   2x1 = 2x2 = + 2k + 2k

7. : 2 c) sen 2x = 2 – 2 cos 2x 2 senx cosx = 2 (1 – cos2x) [1 – (1 – 2sen2x)] senx cosx = [1 – 1 + 2sen2x)] senx cosx = 2 sen2x senx cosx = 2 sen2x – senx cosx = 0 senx (2 senx – cosx) = 0

8.  5  sen x = 5  1  sen x = 5 senx (2 senx – cosx)= 0 ó 2 senx – cosx = 0 sen x = 0 x = 1800k 2 senx = cosx kZ 4sen2x = cos2x 4sen2x = 1 – sen2x 5sen2x = 1

9.  5  sen x = 5 TABLA sen x =  0,448 x1 = 26,60 + 3600k x2 = (1800 – 26,60)+ 3600k x2 = 153,40 + 3600k x3 = (1800 + 26,60)+ 3600k x3 = 206,60 + 3600k x4 = (3600 – 26,60)+ 3600k k  Z x4 = 333,40 + 3600k

10. 0,4478

11. Para el estudio individual 5 3 cos2x + sen2x 1 – senx = Resuelve las siguientes ecuaciones con la condición0o  x  360o . a)sen2x + cosx = 0 x = 90o; 270o; 210o; 330o b)cos2x + cos2x = 5sen2x x = 30o; 150o ;210o ;330o c)cos2x + cosx = 0 x=60o; 300o; 180o d) x = 90o; 41,8o; 138,2o