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1) To find a general solution, first use algebra to transform the equation into the form:
\n" ); document.write( "trigfunction(something) = number
\n" ); document.write( "3 = 2sin (3(x+ (pi/3)) +4
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\n" ); document.write( "\n" ); document.write( "Subtracting 4 from each side we get:
\n" ); document.write( "-1 = 2sin (3(x+ (pi/3))
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\n" ); document.write( "\n" ); document.write( "Dividing both sides by 2 we get:
\n" ); document.write( "-1/2 = sin (3(x+ (pi/3))
\n" ); document.write( "which is the desired form.
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\n" ); document.write( "\n" ); document.write( "2) Next, we need to find expressions for angles whose sin's are -1/2. Since sin(pi/6) = 1/2 the reference angle is pi/6. And since sin is negative in the 3rd and 4th quadrants, we are looking for angles in those quadrants with a reference of pi/6.
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\n" ); document.write( "\n" ); document.write( "In the third quadrant our expression would be:
\n" ); document.write( "pi + pi/6 + 2pi*n
\n" ); document.write( "which simplifies to:
\n" ); document.write( "7pi/6 + 2pi*n
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\n" ); document.write( "\n" ); document.write( "In the fourth quadrant our expression would be:
\n" ); document.write( "2pi - pi/6 + 2pi*n
\n" ); document.write( "which simplifies to:
\n" ); document.write( "11pi/6 + 2pi*n
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\n" ); document.write( "\n" ); document.write( "NOTE: The 2pi*n is a way for us to include all the coterminal angles (which would also all have sin values of -1/2. The \"n\" can be replace by any integer. Each different integer results in another angle whose sin is -1/2.
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\n" ); document.write( "\n" ); document.write( "3) Write equations to set the trig function's argument to the expressions from step 2:
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\n" ); document.write( "\n" ); document.write( "3(x+ (pi/3)) = 7pi/6 + 2pi*n
\n" ); document.write( "which simplifies to
\n" ); document.write( "3x + pi = 7pi/6 + 2pi*n
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\n" ); document.write( "\n" ); document.write( "3(x+ (pi/3)) = 11pi/6 + 2pi*n
\n" ); document.write( "which simplifies to
\n" ); document.write( "3x + pi = 11pi/6 + 2pi*n
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\n" ); document.write( "\n" ); document.write( "4) Solve these equations for x.
\n" ); document.write( "Subtracting pi from each side of both equations:
\n" ); document.write( "3x = pi/6 + 2pi*n
\n" ); document.write( "3x = 5pi/6 + 2pi*n
\n" ); document.write( "Multiplying each side by 1/3 (or dividing both sides by 3) we get:
\n" ); document.write( "x = pi/18 + (2/3)pi*n
\n" ); document.write( "x = 5pi/18 + (2/3)pi*n
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\n" ); document.write( "\n" ); document.write( "These last equations are the \"general solution\". \n" ); document.write( "