Download If you know the measure of an angle, you also know its quadrant of

Honors Geometry
Advanced Trigonometry
7. Solving Basic Trig Equations
To solve an equation such as
sin  = -0.4325
You first find the quadrants of
termination and the reference angle.
Sine is negative is Quads III and IV.
sin-1(0.4325) = 25.6
(We use 0.4325 instead of –0.4325
because the negative is already
addressed by identifying the quadrants.)
So the reference angle, , is 25.6
In Quadrant III,
 =  + 180
= 25.6 + 180
= 205.6
In Quadrant IV,
 = 360 - 
= 360 - 25.6
= 334.4
solution : { 205.6, 334.4 }
Examples
cos  = 0.75
Ref.  : 41.4
I
IV
 = 
 = 360 - 
= 41.4
= 360 – 41.4
= 318.2
solution : { 41.4, 318.2 }
tan  = 3
Ref.  : 71.6
I
III
 = 
 =  + 180
= 71.6
= 71.6 + 180
= 251.6
solution : { 71.6, 251.6 }
cos  = -0.8742
Ref.  : 29.0
II
 = 180 - 
= 180 – 29.0
= 151.0
III
 =  + 180
= 29.0 + 180
= 209.0
solution : { 151.0, 209.0 }
sin  = 1.248
Since –1 < sin  < 1,
The equation can’t be true.
solution : 
5∙tan  + 8 = 2
Solve for tan .
5∙tan  = -6
tan  = -1.2
Ref.  : 50.2
II
IV
 = 180 - 
 = 360 - 
= 180 - 50.2
= 360 - 50.2
= 129.8
= 309.8
solution : { 129.8, 309.8 }
-8∙cos  - 5 = -5
Solve for cos .
-8∙cos  = 0
cos  = 0
 must be a quadrant angle, since only
quadrant angles have cosines (or sines)
equal to 0, 1, or –1.
Cosine (the x value on the unit circle) is
zero at 90 and 270.
solution : { 90, 270 }
Honors Geometry
Advanced Trigonometry
7. Solving Basic Trig Equations
Exercises
1. sin  = 0.3465
25. 2∙sin  = 0.3468
2. cos  = 0.9423
26. cos  + 1 = 0.9423
3. tan  = 3.567
27. tan  - 7 = 3.567
4. sin  = -0.6529
28. -5∙sin  = -0.6525
5. cos  = -0.3478
29. 2∙cos  - 2 = -0.3578
6. tan  = -0.1289
30. 4∙tan  + 2 = 2.678
7. sin  = 0.8935
31. -sin  + 1 = 1.9578
8. cos  = 0.2954
32. 3∙cos  - 2 = 0.6667
9. tan  = 6
33. -8∙tan  - 5 = -3
10. sin  = 1.239
34. 4∙sin  - 3 = 0.3456
11. cos  = -3.003
35. cos  - 2 = -0.55
12. tan  = 2.5
36. ½∙tan  = 1.5
13. sin  = 5/12
37. -4∙sin  = 2/3
14. cos  = -3/11
38. 2∙cos  = 5/13
15. tan  = 17/7
39. 5∙tan  = 17/4
16. sin  = -4/13
40. 3∙sin  = -3/8
17. cos  = -55/67
41. 10∙cos  = -5/9
18. tan  = -1
42. -tan  - 5 = -1/7
19. sin  = -0.5
43. ½ sin  = -4/11
20. cos  = 0.5
44. –0.25∙cos  = -5/28
21. tan  =
3
45. 11∙tan  = -66/235
22. sin  =
2
2
23. cos  = 24. tan  = 0
3
2
46. ⅓∙sin  = 1/15
47. ⅜∙cos  = 9/25
48. -⅞∙tan  = 259/32