Question 858136: What are the values that satisfy the trigonometric equation for 0 < q < 2
sin(θ)+tan(-θ)=0
Found 2 solutions by jim_thompson5910, Theo:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I'm going to use x in place of theta
sin(x)+tan(-x)=0
sin(x)-tan(x)=0 ... since tan(x) is odd
sin(x)-sin(x)/cos(x)=0
sin(x)*cos(x)/cos(x)-sin(x)/cos(x)=0
( sin(x)*cos(x)-sin(x) )/cos(x)=0
sin(x)*cos(x)-sin(x) = 0*cos(x)
sin(x)*cos(x)-sin(x) = 0
sin(x)( cos(x)-1 ) = 0
sin(x) = 0 or cos(x)-1 = 0
sin(x) = 0 or cos(x) = 1
x = arcsin(0) or x = arccos(1)
x = 0, pi or x = 0
So if x is in the interval [0, 2pi), then the solutions are: x = 0, x = pi
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! as far as i can tell, the solution appears to be x = 0, or x = pi.
you can graph this equation which will show you the zero points.
the vertical lines at x = 0 and x = 2 are the left and right limits of the valid range.
you can see there is a zero crossing at x = 0 and x = pi.
pi is equal to approximately 3.14.
algebraically, you would solve it as follows:
start with:
sin(x) + tan(-x) = 0
since tan(-x) is equal to -tan(x), you can rewrite as follows:
sin(x) - tan(x) = 0
since tan(x) is equal to sin(x) / cos(x), replace tan(x) with that to get:
sin(x) - sin(x)/cos(x) = 0
take sin(x) by itself and multiply it by cos(x) / cos(x).
the equation becomes:
sin(x)cos(x)/cos(x) - sin(x)/cos(x) = 0
you can now combine the numerators under the common denominator of cos(x) to get:
(sin(x)cos(x) - sin(x))/cos(x) = 0
factor the numerator to get:
sin(x)(cos(x)-1) = 0
this is true if sin(x) = 0 and if cos(x)-1 = 0
you get 2 solutions to this equation.
they are:
sin(x) = 0
cos(x) = 1
the equation of sin(x) + tan(-x) = 0 is true when sin(x) = 0 and when cos(x) = 1.
sin(x) = 0 when x = -2pi, -pi, 0, pi, 2pi.
cos(x) = 1 when x = -2pi, 0, 2pi.
the graph of cosine (x) and sin(x) is shown below:
you can see that cosine(x) = 1 when x is equal to { -2pi, 0, 2pi } and sin(x) is equal to 0 when x is equal to { -2pi, -pi, 0, pi, 2pi }.
pi is equal to approximately 3.14 and 2pi is equal to approximately 6.28.
in the interval between 0 and 2, the only time when the equation of sin(x) + tan(-x) is equal to 0 is when x = 0. -pi and pi are outside of the range.
the vertical lines at x = 0 and x = 2 are the left and right limits of the valid range.
the graph of the equation of sin(x) + tan(-x) is shown below again for your convenience.
the vertical lines at x = 0 and x = 2 are the left and right limits of the valid range.
you can see there is a zero crossing at x = 0 and x = pi.
0 is within the interval of x = 0 to 2.
pi is not.
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