SOLUTION: Find all solutions of each of the equations in the interval [0,2pi).
a) sin(x+pi/3)+sin(x−pi/3)=1
b) tan(x+pi)+2sin(x+pi)=0
c) cos(x−pi/2)+sin2x=0
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Question 1077821: Find all solutions of each of the equations in the interval [0,2pi).
a) sin(x+pi/3)+sin(x−pi/3)=1
b) tan(x+pi)+2sin(x+pi)=0
c) cos(x−pi/2)+sin2x=0
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
a)
Using the trigonometric identities
for sine of sum an difference of two angles,
the equation can be re-written as
Taking out and as common factors
the equation can be re-written as
We know that , so we re-write the equation as
and .
In the interval [0,2pi), that happens only for
.
b)
Based on trigonometric identities, the equation can be re-written as
and .
Then, with some algebra, it can be re-written as
and
The numerator is zero when
---> or .
The numerator is also zero when
---> ---> or .
For none of those values of x, is the zero,
so they are all valid solutions.
c)
(Or did you mean instead?)
Using trigonometric identities,
the equation can be re-written as
<--> and .
If the second term was really AMP Parsing Error of []: Invalid expression '': syntax error at /home/ichudov/project_locations/algebra.com/templates/Algebra/Expression.pm line 203.
.
,
using the trig identity for double angles,
the equation can be re-written as
<--->
The expression is zero when
---> or .
The expression is also zero when
<---> .
In the interval [0,2pi), that happens for
or .
NOTE: For ,
<--> ,
in the interval [0,2pi) has solutions when
--> or ,
and when
--> --> .
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