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Honors Geometry
Advanced Trigonometry
8. Solving Trig Equations by Factoring
Some more complicated trig equations can
be solved through factoring.
We use the principle from Algebra :
If a∙b = 0, then a = 0 or b = 0
The equation
tan2 - 5tan + 6 = 0
can be factored to
(tan - 2)(tan - 3) = 0
Separating,
tan - 2 = 0
tan = 2
or
or
tan - 3 = 0
tan = 3
tan = 2
ref  : 63.4
I :  = 63.4
III :  = 180 + 63.4 = 243.4
tan = 3
ref  : 71.6
I :  = 71.6
III :  = 180 + 71.6 = 251.6
Solution : {63.4, 71.6, 243.4, 251.6}
Example
2sin3 + sin2 - sin = 0
Factoring,
sin(2sin2 + sin - 1) = 0
sin(2sin - 1)(sin + 1) = 0
Separating,
sin = 0 or 2sin - 1 = 0 or sin + 1 = 0
sin = 0 or sin = ½ or sin = -1
sin = 0
quadrant angle.  = 0 or 180
sin = ½
ref  : 30
I :  = 30
II :  = 180 – 30 = 150
sin = -1
quadrant angle.  = 270
Solution : {0, 30, 150, 180, 270}
Some trig equations with more than one trig
function can also be solved by factoring.
The equation
sin∙tan + sin = 0
can be factored to
sin(tan + 1) = 0
Separating,
sin = 0 or tan + 1 = 0
sin = 0 or tan = -1
sin = 0
quadrant angle.  = 0 or 180
tan = -1
ref  : 45
II :  = 180 – 45 = 135
IV :  = 360 – 45 = 315
Solution : {0, 135, 180, 315}
Example
4sin2∙cos - 9sin∙cos + 2cos = 0
Factoring,
cos(4sin2 - 9sin + 2) = 0
cos(4sin - 1)(sin - 2) = 0
Separating,
cos = 0 or 4sin - 1 = 0 or sin - 2 = 0
cos = 0 or sin = 0.25 or sin = 2
cos = 0
quadrant angle.  = 90 or 270
sin = 0.25
ref  : 14.5
I :  = 14.5
II :  = 180 – 14.5 = 165.5
sin = 2
impossible since –1 < sin < 1
Solution : {14.5, 90, 165.5, 270}
Honors Geometry
Advanced Trigonometry
8. Solving Trig Equations by Factoring
1. 2sin2 + sin  - 1 = 0
16. sin∙cos = 0
2. 20cos2 + 33cos + 10 = 0
17. tan∙sin2 - tan = 0
3. tan2 - 4tan = 0
18. cos∙tan2 - 2cos∙tan - 3cos = 0
4. 40sin2 + 38sin + 7 = 0
19. 4tan∙sin - 3tan = 0
5. 2cos2 - cos – 1 = 0
20. 8sin2∙cos - 2sin∙cos - cos = 0
6. tan2 - 3tan – 18 = 0
21. sin∙tan2 - 9sin = 0
7. 25sin2 - 9 = 0
22. 5sin∙cos2 - 9sin∙cos + 4sin = 0
8. 2cos2 - 1 = 0
23. tan - 4 sin2∙tan = 0
9. tan2 - 3 = 0
24. 8sin2∙cos - 10sin∙cos + 3 cos = 0
10. 18sin2 = 9sin + 2
25. 2tan∙cos2 + tan∙cos = tan
11. 56cos2 + 15 = 61cos
26. cos∙tan2 = cos∙tan + 2cos
12. 13tan - 36 = tan2
27. 2sin2∙cos + cos = 3sin∙cos
13. 2sin3 - 3sin2 - 2sin = 0
14. cos3 - 2cos2 - 3cos = 0
15. tan4 +7tan3 + 10tan2 = 0