document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=806203>Question 806203</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Any help you could give on this problem would be great. Thank you!!!\r<BR>\n" );
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document.write( "For 0 degrees &#8804; theta < 1080 degrees, find all values of theta such that cos theta = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++%28-sqrt+%283%29+%2F+2%29+ ALIGN=MIDDLE  ALT=\"++%28-sqrt+%283%29+%2F+2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "I have discovered that all possible answers have to be in quadrant 2 (I THINK), because that is where the cosine is negative, but not both numerator and denominator of the number have to be negative (like they would in quadrant three). Knowing that cos theta is x/r, this allows you to know the x and r values, of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-sqrt+%283%29+ ALIGN=MIDDLE  ALT=\"+-sqrt+%283%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and 2, respectively. Then you can put them in the formula for r=2, of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt+%28x%5E2+%2By%5E2%29+ ALIGN=MIDDLE  ALT=\"+sqrt+%28x%5E2+%2By%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to find y as 1. I know that since 1080/360 is 3, there are three values, but this is as much information as I have been able to find. I am struggling to find the degree value for theta, and transfer the radians into something usable (if that makes sense). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #485780 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=485780\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are some values for sine and cosine that are easy to remember (no memorization required), and it is easy to remember the first quadrant angles they correspond to.<BR>\n" );
document.write( "The values are 0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and 1.<BR>\n" );
document.write( "I'll explain later why they are easy to remember (at least for me).<BR>\n" );
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document.write( "We figure out problems for a \"reference angle\" in the first quadrant, and then \"translate\" the findings to other quadrants as needed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%5Eo%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%2830%5Eo%29=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a known reference angle and trigonometric function value.<BR>\n" );
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document.write( "Cosine has the same sign as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in all quadrants, so negative cosines are found in quadrants II and III.<BR>\n" );
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document.write( "Angles in quadrant II have the same sine and opposite cosine as their supplements, who are in quadrant I.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28150%5Eo%29=cos%28180%5Eo-30%5Eo%29=-cos%2830%5Eo%29=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%28150%5Eo%29=cos%28180%5Eo-30%5Eo%29=-cos%2830%5Eo%29=-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "Angles in quadrant III have the opposite sine and cosine as their <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > smaller first quadrant reference angles.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28210%5Eo%29=cos%28180%5Eo%2B30%5Eo%29=-cos%2830%5Eo%29=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%28210%5Eo%29=cos%28180%5Eo%2B30%5Eo%29=-cos%2830%5Eo%29=-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "So far we have found the angles that have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%28theta%29=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the first turn, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=theta%3C360%5Eo ALIGN=MIDDLE  ALT=\"0%3C=theta%3C360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are no answers in quadrants I and IV, where cosine is positive, and the values for cosine (or for any other trig function) do not repeat within a quadrant, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=150%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=150%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=210%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=210%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the only solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=theta%3C360%5Eo ALIGN=MIDDLE  ALT=\"0%3C=theta%3C360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "As you said, there are 3 whole <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > turns in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1080%5Eo ALIGN=MIDDLE  ALT=\"1080%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and since cosine has a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the same cosine values are repeated for angles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=720%5Eo ALIGN=MIDDLE  ALT=\"720%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > larger.<BR>\n" );
document.write( "That gives us 4 more answers:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=150%5Eo%2B360%5Eo=highlight%28510%5Eo%29 ALIGN=MIDDLE  ALT=\"theta=150%5Eo%2B360%5Eo=highlight%28510%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=210%5Eo%2B360%5Eo=highlight%28570%5Eo%29 ALIGN=MIDDLE  ALT=\"theta=210%5Eo%2B360%5Eo=highlight%28570%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=150%5Eo%2B720%5Eo=highlight%28870%5Eo%29 ALIGN=MIDDLE  ALT=\"theta=150%5Eo%2B720%5Eo=highlight%28870%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=210%5Eo%2B720%5Eo=highlight%28930%5Eo%29 ALIGN=MIDDLE  ALT=\"theta=210%5Eo%2B720%5Eo=highlight%28930%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You could express all the answers as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%282k-1%29%2A180%5Eo+%2B-+30%5Eo ALIGN=MIDDLE  ALT=\"theta=%282k-1%29%2A180%5Eo+%2B-+30%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > an element of {1, 2, 3} , if you wanted to look fancy. (That is kind of formula is what you have to use if they ask you for all the angles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, no matter how large).<BR>\n" );
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document.write( "HOW TO KNOW THE VALUES LISTED AT THE TOP:<BR>\n" );
document.write( "Below is an equilateral triangle with sides measuring 1. Drawing an altitude, we split that triangle symmetrically into two congruent right triangles.<BR>\n" );
document.write( "The angles in the equilateral triangle measure <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the altitute split one of them into two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%5Eo ALIGN=MIDDLE  ALT=\"30%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles.<BR>\n" );
document.write( "The hypotenuse of one of those right triangles measures 1, because it is the side of the equilateral triangle.<BR>\n" );
document.write( "The short leg of one of the right triangles measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because it is half of a side of the equilateral triangle.<BR>\n" );
document.write( "According to the Pythagorean theorem, the long leg measures<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%5E2-%281%2F2%29%5E2%29=sqrt%281=1%2F4%29=sqrt%283%2F4%29=sqrt%283%29%2Fsqrt%284%29=sqrt%283%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > length of the short leg, opposite the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%5Eo ALIGN=MIDDLE  ALT=\"30%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle and adjacent to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%5Eo%29=cos%2860%5Eo%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%2830%5Eo%29=cos%2860%5Eo%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > length of the long leg, opposite the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle and adjacent to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%5Eo ALIGN=MIDDLE  ALT=\"30%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28600%5Eo%29=cos%2830%5Eo%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28600%5Eo%29=cos%2830%5Eo%29=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.2%2C1.2%2C-0.4%2C2%2C%0D%0Atriangle%28-1%2C0%2C1%2C0%2C0%2C1.414%29%2Cline%280%2C0%2C0%2C1.414%29%2C%0D%0Arectangle%280%2C0%2C0.1%2C0.1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > No need to memorize without understanding, and you can recalculate those values if you forget them.<BR>\n" );
document.write( "Now, consider a square with side length 1, and its diagonal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C1.1%2C-0.1%2C1.1%2C%0D%0Arectangle%280%2C0%2C1%2C1%29%2Cline%280%2C0%2C1%2C1%29%2C%0D%0Arectangle%280.95%2C0%2C1%2C0.05%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The diagonal splits the square into two isosceles right triangles, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles opposite legs og length 1.<BR>\n" );
document.write( "According to the Pythagorean theorem, the hypotenuse measures<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%5E2%2B1%5E2%29=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%281%5E2%2B1%5E2%29=sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the value for sine and cosine of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%5Eo ALIGN=MIDDLE  ALT=\"45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles are the ratio of leg to hypotenuse:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282%29=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282%29=sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "MORE:<BR>\n" );
document.write( "Trigonometric functions are often defined as the coordinates of a point determined by the angle in \"standard position\" and the \"unit circle:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.2%2C1.2%2C-1.2%2C1.2%2C%0D%0Agrid%280%29%2Ccircle%280%2C0%2C1%29%2C%0D%0Aarrow%280%2C0%2C1.125%2C0.65%29%2Carrow%280%2C0%2C-1.125%2C+0.65%29%2C%0D%0Ablue%28circle%281%2C0%2C0.05%29%29%2C%0D%0Ablue%28circle%280.866%2C0.5%2C0.05%29%29%2Cblue%28circle%28-0.866%2C0.5%2C0.05%29%29%2C%0D%0Aarc%280%2C0%2C0.3%2C0.3%2C-150%2C0%29%2C+arc%280%2C0%2C0.5%2C0.5%2C-30%2C0%29%2C%0D%0Alocate%280.25%2C0.2%2C30%5Eo%29%2Clocate%28-0.28%2C0.3%2C150%5Eo%29%2C%0D%0Alocate%280.02%2C-0.02%2CO%29%2Clocate%281.05%2C-0.05%2CA%29%0D%0Alocate%280.87%2C0.55%2CB%29%2Clocate%28-0.8%2C0.58%2CC%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%5Eo%29=cos%28AOB%29=x%5BB%5D=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%2830%5Eo%29=cos%28AOB%29=x%5BB%5D=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%5Eo%29=y%5BB%5D=1%2F2 ALIGN=MIDDLE  ALT=\"sin%2830%5Eo%29=y%5BB%5D=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the coordinates of point B.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28150%5Eo%29=cos%28AOC%29 ALIGN=MIDDLE  ALT=\"cos%28150%5Eo%29=cos%28AOC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28150%5Eo%29=sin%28AOC%29 ALIGN=MIDDLE  ALT=\"sin%28150%5Eo%29=sin%28AOC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the coordinates of point C </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );