| Easy as 2, 3, 4 |
Find all x between 0 and 90 degrees that satisfy the equation
sin(2x) + tan(3x) + cos(4x) = 2 |
Problem Moderated by: Douglas |
| Problem Solution |
It would be nice if there weren't a trigonometric function of 3x in this equation -- if you had only to solve sin(2x)+cos(4x)=1, for example. Unfortunately, this was not the case, and it becomes necessary to reduce the equation to terms of x only. Begin by multiplying through by cos(3x):
cos(3x)sin(2x)+sin(3x)+cos(3x)cos(4x)-2cos(3x)=0
Some quick theorems:
sin(2x)=2sin(x)cos(x)
cos(2x)=1-2sin2(x)
cos(3x)=cos(2x+x)=cos(2x)cos(x)-sin(2x)sin(x)=
cos(x)-2cos(x)sin2(x)-2sin2(x)cos(x)
sin(3x)=sin(2x+x)=sin(2x)cos(x)+cos(2x)sin(x)=
2cos2(x)sin(x)+sin(x)-2sin3(x)
cos(4x)=cos(2(2x))=1-2sin2(2x)=
1-8sin2(x)cos2(x)
x= 15 or 75 |
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