document.write( "Question 858136: What are the values that satisfy the trigonometric equation for 0 < q < 2
\n" ); document.write( "sin(θ)+tan(-θ)=0 \n" ); document.write( "

Algebra.Com's Answer #517058 by Theo(13342) $\"\"$ $\"About$

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as far as i can tell, the solution appears to be x = 0, or x = pi.
\n" ); document.write( "you can graph this equation which will show you the zero points.
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\n" ); document.write( "the vertical lines at x = 0 and x = 2 are the left and right limits of the valid range.
\n" ); document.write( "you can see there is a zero crossing at x = 0 and x = pi.
\n" ); document.write( "pi is equal to approximately 3.14.
\n" ); document.write( "algebraically, you would solve it as follows:
\n" ); document.write( "start with:
\n" ); document.write( "sin(x) + tan(-x) = 0
\n" ); document.write( "since tan(-x) is equal to -tan(x), you can rewrite as follows:
\n" ); document.write( "sin(x) - tan(x) = 0
\n" ); document.write( "since tan(x) is equal to sin(x) / cos(x), replace tan(x) with that to get:
\n" ); document.write( "sin(x) - sin(x)/cos(x) = 0
\n" ); document.write( "take sin(x) by itself and multiply it by cos(x) / cos(x).
\n" ); document.write( "the equation becomes:
\n" ); document.write( "sin(x)cos(x)/cos(x) - sin(x)/cos(x) = 0
\n" ); document.write( "you can now combine the numerators under the common denominator of cos(x) to get:
\n" ); document.write( "(sin(x)cos(x) - sin(x))/cos(x) = 0
\n" ); document.write( "factor the numerator to get:
\n" ); document.write( "sin(x)(cos(x)-1) = 0
\n" ); document.write( "this is true if sin(x) = 0 and if cos(x)-1 = 0
\n" ); document.write( "you get 2 solutions to this equation.
\n" ); document.write( "they are:
\n" ); document.write( "sin(x) = 0
\n" ); document.write( "cos(x) = 1
\n" ); document.write( "the equation of sin(x) + tan(-x) = 0 is true when sin(x) = 0 and when cos(x) = 1.
\n" ); document.write( "sin(x) = 0 when x = -2pi, -pi, 0, pi, 2pi.
\n" ); document.write( "cos(x) = 1 when x = -2pi, 0, 2pi.
\n" ); document.write( "the graph of cosine (x) and sin(x) is shown below:
\n" ); document.write( " $\"graph%28600%2C600%2C-7%2C7%2C-2%2C2%2Ccos%28x%29%2Csin%28x%29%2C50%28x%29%2C50%28x-2%29%29\"$
\n" ); document.write( "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 }.
\n" ); document.write( "pi is equal to approximately 3.14 and 2pi is equal to approximately 6.28.
\n" ); document.write( "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.
\n" ); document.write( "the vertical lines at x = 0 and x = 2 are the left and right limits of the valid range.
\n" ); document.write( "the graph of the equation of sin(x) + tan(-x) is shown below again for your convenience.
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\n" ); document.write( "the vertical lines at x = 0 and x = 2 are the left and right limits of the valid range.
\n" ); document.write( "you can see there is a zero crossing at x = 0 and x = pi.
\n" ); document.write( "0 is within the interval of x = 0 to 2.
\n" ); document.write( "pi is not.
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