1. Trigonometric Equations Edited by Mr. Francis Hung Last Updated: 2008-12-04

2. Trigonometric Equations sin x = sin  x =  or 180  -  sin x = sin 30  x = 30  or 180  - 30  x = 30  or 150  sin x = sin (-120  ) x = -120  or 180  -(-120  ) or -120  +360  x = 300  or 240 

3. sin x = sin  then x =  or 180  -  sin x = -1 x = -90  or 180  - (-90  ) x = 270 

4. sin x = sin  then x =  or  -  sin x = 1.2  -1  sin x  1  x has no solution

5. cos x = cos 130  x = 130  or 360  - (130  ) x = 130  or 230  cos x = cos  then x =  or 360  -  cos x = -0.9 x = 154  or 360  - 154  x = 154  or 206  cos x = -3  -1  cos x  1  x has no solution

6. cos x = cos  then x =  or 360  -  cos x = cos (-20  ) cos x = cos 20  x = 20  or 360  - 20  x = 20  or 340  cos x = cos (-10  ) x = -10  or 360  - (-10  ) or 360  + (-10  ) or 10  x = 10  or 350 

7. cos x = cos  then x =  or 2  -  cos x = tan 0.5 c cos x = 0.5463 x = 0.9929 or 2  - 0.9929 x = 0.993 or 5.29

8. tan x = tan  then x =  or 180  +  tan x = -1 x = -45  or 180  + (-45  ) or 360  + (-45  ) x = 135  or 315  tan x = 5 x = 78.7  or 259 

9. tan x = tan  then x =  or 180  +  sin x = -2cos x tan x = -2 x = -63.4  or 180  + (-63.4  ) or 360  + (-63.4  ) x = 117  or 297  tan x = -2 (sin 60  + 1) tan x = -3.73 x = 105  or 285 

10. tan x = tan  then x =  or  +  tan x = -0.5 x = -0.464 c or  - 0.464 c or 2  - 0.464 c x = 2.68 or 5.82

11. Exercise: solve the trigonometric equations 1.sin x = sin(-15  ) 195  or 345  2.Answer in radians: sin x = 0.6 0.644 or 2.50 3.Answer in terms of  : 4.sin x = 7 no solution 5.cos x = cos -330  30  or 330  6.cos x = 0 x = 90  or 270  7.Answer in radians: cos x = -1/3 1.91 or 4.37

12. Exercise: solve the trigonometric equations 8.Answer in terms of  : cos x = -1  9.Answer in terms of  : cos x = -sin(3  /4) 3  /4 or 5  /4 10.Answer in terms of  : 11.tan x = tan 540  0 , 180  or 360  12.3 sin x = 2 cos x 33.7  or 214  13.Answer in terms of  : tan x = -1 x = 3  /4 or 7  /4 14.Answer in radians: tan x = 3 1.25 or 4.39

13. More difficult examples 1.cos 2x = cos 60  2x = 60 , 300 , 420 , 660  x = 30 , 150 , 210 , 330 

14. More difficult examples 3.cos 2x = cos (10  + x) 2x = 10  + x or 2x = 360  - (10  + x) x = 10  or 116.67  Is there any other solution between 0  and 360  ? 236.67 , 356.67  4.2 cos 2  - 3 cos  + 1 = 0 (2 cos  - 1)(cos  - 1) = 0 cos  = 0.5 or cos  = 1  = 60 , 300  or 0 , 360 

15. More difficult examples 5.2 tan 2  + tan  - 1 = 0 (Answer in radians.) (2 tan  - 1)(tan  + 1) = 0 tan  = 0.5 or tan  = -1  = 0.464 c, 3.61 c or 3  /4, 7  /4 6.cos 3x = sin 2x cos 3x = cos(90  - 2x) 3x = 90  - 2x or 3x = 360  - (90  - 2x) x = 18  or 270  Is there any other solution between 0  and 360  ? 90 , 162 , 234 , 306 

16. More difficult examples 7.2 sin 2  - cos  - 1 = 0 (Answer in terms of .) 2(1- cos 2  ) - cos  - 1 = 0 2 cos 2  + cos  - 1 = 0 (2 cos  - 1)(cos  + 1) = 0 cos  = 0.5 or cos  = -1  =  /3, 5  /3 or  8.sin  tan  + cos  = 1 (Answer in terms of .) sin  ( sin  / cos  ) + cos  = 1 sin 2  + cos 2  = cos  cos  = 1  = 0 c or 2 

17. More difficult examples 9.3 - 2 sin  cos  - 4 sin 2  = 0 3(sin 2  + cos 2  ) - 2 sin  cos  - 4 sin 2  = 0 3 cos 2  - 2 sin  cos  - sin 2  = 0 3 - 2 tan  - tan 2  = 0 tan 2  + 2 tan  - 3 = 0 (tan  + 3)(tan  - 1) = 0 tan  = -3 or tan  = 1  = 108 , 288  or 45 , 225 

18. More difficult examples 10.12 sin  - 5 cos  = 13 (answer in radians.) (12 sin  - 5 cos  ) 2 = 169 144 sin 2  -120sin  cos  +25cos 2  =169(sin 2  +cos 2  ) 25 sin 2  + 120 sin  cos  + 144 cos 2  = 0 25 tan 2  + 120 tan  + 144 = 0 (5 tan  + 12) 2 = 0 tan  = -12/5  = 1.97 c, 5.11 c Check: when  = 1.97 c, LHS = 12 sin 1.97 c - 5 cos 1.97 c = 13 = RHS when  = 5.11 c, LHS = 12 sin 5.11 c - 5 cos 5.11 c = -13  RHS   = 1.97 c only

19. Summary In degrees, sin x = sin  then x =  or 180  -  cos x = cos  then x =  or 360  -  tan x = tan  then x =  or 180  +  In radians, sin x = sin  then x =  or  -  cos x = cos  then x =  or 2  -  tan x = tan  then x =  or  + 