document.write( "Question 696450: Solve for all real numbers. Base your answers on the unit circle and fundamental period accordingly.
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\n" ); document.write( "2sin(2θ)=3 \r
\n" ); document.write( "\n" ); document.write( "The answer is: (pi/6, n pi, pi/3 + n pi) \r
\n" ); document.write( "\n" ); document.write( "I don't know what it's asking me to do, so I have no idea how to go about even starting the problem. Please try to help! Thank you! \n" ); document.write( "
Algebra.Com's Answer #429123 by KMST(5328)\"\" \"About 
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Something got lost in transcription here,
\n" ); document.write( "\"2sin%282theta%29=3\" <--> \"sin%282theta%29=3%2F2\" cannot be,
\n" ); document.write( "because \"sin%28anything%29%3C=1\" and \"3%2F2%3E1\"
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\n" ); document.write( "The problem should say \"2sin%282theta%29=sqrt%283%29\"
\n" ); document.write( "\"2sin%282theta%29=sqrt%283%29\" --> \"sin%282theta%29=sqrt%283%29%2F2\"
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\n" ); document.write( "Here is the unit circle with the angles that have a sine of \"sqrt%283%29%2F2\"
\n" ); document.write( " The unit circle is a circle of radius 1, centered at the origin of a x-y coordinate system.
\n" ); document.write( "Angles on the unit circle are turns or sweeps starting at ray OA.
\n" ); document.write( "Counterclockwise turns are positive, clockwise turns are negative angles.
\n" ); document.write( "You can go more than one turn, as in \"turn the knob \"540%5Eo\"\", meaning one and a half turns counterclockwise.
\n" ); document.write( "Angle AOB measures \"60%5Eo\", \"1%2F6\" of a turn,
\n" ); document.write( "but we prefer to measure it in terms of the arc length from A to B,
\n" ); document.write( "which is \"1%2F6\" of the \"2pi\" length of the circumference,
\n" ); document.write( "so it is \"pi%2F3\".
\n" ); document.write( "We say we measure it in \"radians\" meaning it's the length of the arc measured with the radius as a unit.
\n" ); document.write( "We construct right triangle POB, and define the function sine of AOB as the y-coordinate of B, which is the length of segment PB.
\n" ); document.write( "Triangle POB is half of triangle (not drawn, you'll have to imagine it) AOB, which is an equilateral triangle.
\n" ); document.write( "PO is half of AO, so its length is \"0.5=1%2F2\".
\n" ); document.write( "The length of PB is \"1\", of course.
\n" ); document.write( "Applying Pythagoras, we can find that the length of PB is \"sqrt%283%29%2F2\", so
\n" ); document.write( "\"sin%28pi%2F3%29=sqrt%283%29%2F2\".
\n" ); document.write( "The segment OC is the mirror image of OB, and has the same y-coordinate.
\n" ); document.write( "Angle AOC measures \"120%5Eo\" or \"2pi%2F3\" radians,
\n" ); document.write( "and \"sin%282pi%2F3%29=sqrt%283%29%2F2\".
\n" ); document.write( "Other angles that have the same sine differ in a whole number of turns,
\n" ); document.write( "ending in ray OB or ray OC after turning clockwise
\n" ); document.write( "(same as going to that ray counterclockwise, but then taking a full turn clockwise),
\n" ); document.write( "or turning more than one turn in either direction before ending in that ray.
\n" ); document.write( "The function sine repeats itself periodically after one turn (\"2pi\" radians)
\n" ); document.write( "so we say that the fundamental period of the sine function is \"2pi\".
\n" ); document.write( "can write their measure as
\n" ); document.write( "\"pi%2F3%2B2n%2Api\" or \"2pi%2F3%2B2n%2Api\" , where \"n\" is an integer.
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\n" ); document.write( "In conclusion, from \"sin%282theta%29=sqrt%283%29%2F2\" we conclude that
\n" ); document.write( "\"2theta=pi%2F3%2B2n%2Api\" --> \"theta=%281%2F2%29%28pi%2F3%2B2n%2Api%29\" --> \"highlight%28theta=pi%2F6%2Bn%2Api%29\"
\n" ); document.write( "or \"2pi%2F3%2B2n%2Api\" --> \"theta=%281%2F2%29%282pi%2F3%2B2n%2Api%29\" --> \"highlight%28theta=pi%2F3%2Bn%2Api%29\" \n" ); document.write( "
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