document.write( "Question 807759: How can I find the arccos of 1, with all possible answers? \n" ); document.write( "
Algebra.Com's Answer #486539 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
ANSWER:
\n" ); document.write( "The function \"arc\"\"cos%28x%29\" is defined as the angle \"theta\" that satisfies \"cos%28theta%29=x\" and \"0%3C=theta%3C=pi\"
\n" ); document.write( "(and we measure angles in radians, not degrees).
\n" ); document.write( "If the question is finding the angle \"theta\" such that \"cos%28theta%29=1\",
\n" ); document.write( "then there are infinite answers,
\n" ); document.write( "so you have to give them as \"theta+=2k%2Api\" in radians,
\n" ); document.write( "(or \"theta+=k%2A360%5Eo\" if you measure angles in degrees),
\n" ); document.write( "where \"k\" is any integer.
\n" ); document.write( "However, to define \"arc\"\"cos%28x%29\" as a function,
\n" ); document.write( "we choose just the most trivial of those answers, \"theta=0\" ,
\n" ); document.write( "so \"arc\"\"cos%281%29=0\" (just one answer).
\n" ); document.write( "
\n" ); document.write( "EXPLANATION AND MORE:
\n" ); document.write( "The drawing below shows the unit circle and an angle, \"theta=AOB\" , in standard position.
\n" ); document.write( " \"theta+=30%5Eo\" or \"theta=pi%2F6\" depending on the units you use for angles.
\n" ); document.write( "Angles are measured as turns counterclockwise from ray OA to ray OB.
\n" ); document.write( "It is a useful way to define angles for technical issues,
\n" ); document.write( "such as stating the angle a control knob has to be turned to get to the desired position.
\n" ); document.write( "You start with OB coinciding with OA and \"theta=0\".
\n" ); document.write( "As you turn OB around O, you eventually go one full turn and get back to OA,
\n" ); document.write( "and that turn is considered to be \"360%5Eo\" or \"2pi\".
\n" ); document.write( "After another turn, the angle would be \"720%5Eo\" or \"4pi\"
\n" ); document.write( "I say that the angle \"theta\" in my drawing is \"30%5Eo\" ,
\n" ); document.write( "because I say that I just turned from OA to OB the shortest possible way.
\n" ); document.write( "If I said that I turned a little over 2 turns clockwise on my way from OA to OB, then it would be
\n" ); document.write( "\"theta=720%5Eo%2B30%5Eo=750%5Eo\"
\n" ); document.write( "Cosine is defined as the x-coordinate of point B, while sine is the y-coordinate.
\n" ); document.write( "In my drawing, those coordinates are
\n" ); document.write( "\"x%5BB%5D=cos%28theta%29=sqrt%283%29%2F2=about\"\"0.866\" and
\n" ); document.write( "\"y%5BB%5D=sin%28theta%29=1%2F2=0.5\" .
\n" ); document.write( "There is another point in the circle (point C) with the same x-coordinate,
\n" ); document.write( "as there is another point (point D) with the same y-coordinate.
\n" ); document.write( "
\n" ); document.write( "It is obvious that the only point in the circle with \"x=1\" is point A,
\n" ); document.write( "so for \"cos%28theta%29=1\" we get infinite answers, but just one answer per turn,
\n" ); document.write( "and the answers are all \"2pi\" or \"360%5Eo\" apart.
\n" ); document.write( "The same thing happens for \"x=-1\" and \"arc\"\"cos%28-1%29\" at the other extreme of the range of \"x\" values for the circle.
\n" ); document.write( "For values in between those extremes, such as
\n" ); document.write( "\"sqrt%283%29%2F2=about\"\"0.866\" ,
\n" ); document.write( "there are twice as many answers,
\n" ); document.write( "because there are twice as many points with that x-coordinate.
\n" ); document.write( "\"x%5BB%5D=sqrt%283%29%2F2=x%5BC%5D\" so \"cos%28AOB%29=sqrt%283%29%2F2=cos%28AOC%29\"
\n" ); document.write( "and we can say \"AOC=-30%5Eo\" or \"AOC=-pi%2F6\" if we choose the simplest way to define those angles.
\n" ); document.write( "For the infinite answers to \"cos%28theta%29=sqrt%283%29%2F2\" ,
\n" ); document.write( "we have to express all the coterminal angles (that differ from AOB and AOC by a whole number of turns) as
\n" ); document.write( "\"theta=2k%2Api+%2B-+pi%2F6\" (or \"theta=k%2A360%5Eo+%2B-+30%5Eo\" if we use degrees) for any integer \"k\" . \n" ); document.write( "
\n" );