document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207211>Question 1207211</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me solve the following trigonometry exact values with working out:<BR>\n" );
document.write( "sin 2pi/3   <BR>\n" );
document.write( "cos 3pi/4<BR>\n" );
document.write( "tan 5pi/6<BR>\n" );
document.write( "sin 7pi/6<BR>\n" );
document.write( "cos 5pi/4<BR>\n" );
document.write( "tan 4pi/3<BR>\n" );
document.write( "sin 5pi/3<BR>\n" );
document.write( "cos 7pi/4<BR>\n" );
document.write( "tan 11pi/6<BR>\n" );
document.write( "I know it is a lot of questions to ask but I am quite stuck. It would be nice if all of them are complete but if this is too much to ask for atleast half to 3/4s of them would be nice. Thankyou! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844949 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844949\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it is easiest to convert the radians to degrees and work from that.<BR>\n" );
document.write( "it is also easiest to work from reference angles as well.<BR>\n" );
document.write( "it is useful to use the calculator to confirm that you are correct.\r<BR>\n" );
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document.write( "there's a table of trig values for common angles that can help you.<BR>\n" );
document.write( "it is at <a  rel=nofollow HREF= \"https://www.mathematicalway.com/mathematics/trigonometry/trigonometric-ratios/\" target = \"_blank\">https://www.mathematicalway.com/mathematics/trigonometry/trigonometric-ratios/</a>\r<BR>\n" );
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document.write( "this is what the one we will be using look like.\r<BR>\n" );
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document.write( "here are the rules for determining the reference angle.<BR>\n" );
document.write( "the reference angle is the equivalent angle in the first quadrant.\r<BR>\n" );
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document.write( "if the angle is in the first quadrant, the reference angle is equal to that angle.<BR>\n" );
document.write( "if the angle is in the second quadrant, the reference angle is equal to 180 minus that angle.<BR>\n" );
document.write( "if the angle is in the third quadrant, the reference angle is equal to that angle minus 180 degrees.<BR>\n" );
document.write( "if the angle is in the fourth quadrant, the reference angle is equal to 360 minus that angle.\r<BR>\n" );
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document.write( "rule for trig signs are as follows:\r<BR>\n" );
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document.write( "sine is positive in the first and second quadrant, negative in the third and fourth quadrant.<BR>\n" );
document.write( "cosine is positive in the first and fourth quadrant, negative in the second and third quadrant.<BR>\n" );
document.write( "tangent is positive in the first and third quadrant, negative in the second and fourth quadrant.\r<BR>\n" );
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document.write( "first is sin(2pi/3).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 2pi/3*180/pi = 2*180/3 = 120 degrees.<BR>\n" );
document.write( "that angle is in the second quadrant.<BR>\n" );
document.write( "reference angle is 180 - 120 = 60 degrees.<BR>\n" );
document.write( "sine of 60 degrees is sqrt(3)/2.<BR>\n" );
document.write( "sine is positive in the first and second quadrants, so sine of 120 is also equal to sqrt(3)/2.\r<BR>\n" );
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document.write( "to confirm, set your calculator to radians and find sin(2pi/3).<BR>\n" );
document.write( "you will get 2.094395102<BR>\n" );
document.write( "enter 2 * sqrt(3) / 2 in your calculator.<BR>\n" );
document.write( "it will be equal to the same, confirming you are correct.\r<BR>\n" );
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document.write( "second is cos(3pi/4).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 3pi/4 * 180/pi = 3*180/4 = 135 degrees.<BR>\n" );
document.write( "that angle is in the second quadrant.<BR>\n" );
document.write( "equivalent angle in the first quadrant is 180 - 135 = 45 degrees.<BR>\n" );
document.write( "cosine of 45 degrees is equal to sqrt(2)/2.<BR>\n" );
document.write( "cosine is negative in the second quadrant, so cosine(135) degrees is -sqrt(2)/2.\r<BR>\n" );
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document.write( "to confirm, set your calculator to radians and find cosine 135 degrees.<BR>\n" );
document.write( "you will get -.7071067812<BR>\n" );
document.write( "enter -sqrt(2)/2 in your calculator.<BR>\n" );
document.write( "you will get the same, confirming you are correct.\r<BR>\n" );
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document.write( "third is tan (5pi/6).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 5pi/6 * 180/pi = 150 degrees.<BR>\n" );
document.write( "that angle is in the second quadrant.<BR>\n" );
document.write( "reference angle is 180 minus 150 = 30 degrees.<BR>\n" );
document.write( "tan(30) = sqrt(3)/3.<BR>\n" );
document.write( "tangent is positive is negative in the second quadrant.<BR>\n" );
document.write( "tan(150) is equal to -sqrt(3)/3.\r<BR>\n" );
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document.write( "set your calculator to radians.<BR>\n" );
document.write( "tan(5pi/6) = -.5773502692<BR>\n" );
document.write( "-sqrt(3)/3 = the same, confirming you are correct.\r<BR>\n" );
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document.write( "next is sin (7pi/6).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 7pi/6 * 180/pi = 210 degrees.<BR>\n" );
document.write( "210 degrees is in the third quadrant.<BR>\n" );
document.write( "reference angle is 210 minus 180 = 30 degrees.<BR>\n" );
document.write( "sine of 30 degrees is equal to 1/2.<BR>\n" );
document.write( "sine is negative in the third quadrant<BR>\n" );
document.write( "sine of 210 degrees is equal to -1/2.\r<BR>\n" );
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document.write( "to confirm, set your calculator to radians if is not already set to that.<BR>\n" );
document.write( "-1/2 = -.5<BR>\n" );
document.write( "sin(7pi/6) = the same, confirming you are correct.\r<BR>\n" );
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document.write( "next is cos(5pi/4).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 5pi/4 * 180/pi = 5*180/4 = 225 degrees.<BR>\n" );
document.write( "225 is in the third quadrant.<BR>\n" );
document.write( "reference angle is 225 minus 180 = 45 degrees.<BR>\n" );
document.write( "cosine 45 degrees is sqrt(2)/2.<BR>\n" );
document.write( "cosine is negative in the third quadrant.<BR>\n" );
document.write( "cos(5pi/4) is equal to -sqrt(2)/2.\r<BR>\n" );
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document.write( "confirm by using your calculator.<BR>\n" );
document.write( "-sqrt(2)/2 = -.7071067812.<BR>\n" );
document.write( "cos(5pi/4) = the same, confirming you are correct.\r<BR>\n" );
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document.write( "next is tan(4pi/3).<BR>\n" );
document.write( "multiply by 180/pi to get 4pi/3 * 180/p = 4*180/3 = 240.<BR>\n" );
document.write( "240 is in the third quadrant.<BR>\n" );
document.write( "reference angle is 240 minus 180 = 60 degrees.<BR>\n" );
document.write( "tangent 60 degrees is equal to sqrt(3).<BR>\n" );
document.write( "tangent is positive in the third quqadrant.<BR>\n" );
document.write( "tan(4pi/3) = sqrt(3).\r<BR>\n" );
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document.write( "use your calculator to confirm.\r<BR>\n" );
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document.write( "sqrt(3) = 1.732050808.<BR>\n" );
document.write( "tan(4pi/3) = the same, confirming you are correct.\r<BR>\n" );
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document.write( "next is sin(5pi/3).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 5pi/3 * 180/pi = 4*180/3 = 300.<BR>\n" );
document.write( "that angle is in the fourth quadrant.<BR>\n" );
document.write( "reference angle is 360 minus 300 = 60 degrees.<BR>\n" );
document.write( "sine of 60 degrees is sqrt(3)/2.<BR>\n" );
document.write( "since is negative in the fourth quadrant.<BR>\n" );
document.write( "sin(5pi/3) = -sqrt(3)/2.\r<BR>\n" );
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document.write( "use your calculator to confirm.<BR>\n" );
document.write( "-sqrt(3)/2 = -.8660251038.<BR>\n" );
document.write( "sin(5pi/3) = the same, fonriming your are correct.\r<BR>\n" );
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document.write( "next is cos(7pi/4).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 7pi/4 * 180/pi = 7*180/4 = 315 degrees.<BR>\n" );
document.write( "315 degrees is in the fourth quadrant.<BR>\n" );
document.write( "reference angle is 360 minus 315 = 45 degrees.<BR>\n" );
document.write( "cosine(45) degrees is sqrt(2)/2.<BR>\n" );
document.write( "cosine is positive in the fourth quadrant.<BR>\n" );
document.write( "cos(7pi/4) = sqrt(2)/2.\r<BR>\n" );
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document.write( "confirm using your calculator.<BR>\n" );
document.write( "sqrt(2)/2 = .7071067812.<BR>\n" );
document.write( "cos(7pi/4) = the same, fonriming you are correct.\r<BR>\n" );
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document.write( "last is tan(11pi/6).\r<BR>\n" );
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document.write( "multiply by 180/pi to get 11pi/6 * 180/pi = 11*180/6 = 330 degrees.<BR>\n" );
document.write( "330 is in the fourth quadrant.<BR>\n" );
document.write( "reference angle id 360 minus 330 = 30 degrees.<BR>\n" );
document.write( "tan(30) = sqrt(3)/3.<BR>\n" );
document.write( "tangent is negative in the fourth quadrant.<BR>\n" );
document.write( "tan(11pi/6) = -sqrt(3)/3.\r<BR>\n" );
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document.write( "confirm using your calculator.<BR>\n" );
document.write( "-sqrt(3)/3 = -.5773502692.<BR>\n" );
document.write( "tan(11pi/6) = the same, confirming you are correct.\r<BR>\n" );
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document.write( "that should do it.\r<BR>\n" );
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document.write( "since the table also shows the angle in radians, you could also have just stayed with radians.\r<BR>\n" );
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document.write( "you still needed to get the reference angle.\r<BR>\n" );
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document.write( "the rules for finding the reference angle in radians would be.\r<BR>\n" );
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document.write( "if the angle is between 0 and pi(same as between 0 and 90), it's in the first quadrant and the reference angle is equal to the angle.<BR>\n" );
document.write( "if the angle is between pi and 2pi (same as between 90 and 180), it's in the second quadrant and the reference angle is equal to 2pi minus the angle.<BR>\n" );
document.write( "if the angle is between 2pi and 3pi/2 (same as between 180 and 270), it's in the third quadrant and the reference angle is equal to that angle minus 2pi.<BR>\n" );
document.write( "if the angle is between 3pi/2 and 2pi (same as between 270 and 360), it's in the fourth quadrant and the reference angle is equal to 2pi minus the angle.\r<BR>\n" );
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document.write( "i'll use last problem to show you how it works.<BR>\n" );
document.write( "you are looking for tan(11pi/6).<BR>\n" );
document.write( "270 degrees * pi/180 = 270/180 * pi = 3/2 * pi.<BR>\n" );
document.write( "3/2 * pi = 9/6 * pi<BR>\n" );
document.write( "11pi/6 is greater than 9/6 * pi.<BR>\n" );
document.write( "2pi = 12pi/6 is greater than 11pi/6.<BR>\n" );
document.write( "11pi/6 is in the fourth quadrant.<BR>\n" );
document.write( "11pi/6 is in the fourth quadrant.<BR>\n" );
document.write( "the reference angle is 2pi minus 11pi/6 = 12pi/6 minus 11pi/6 = pi/6.<BR>\n" );
document.write( "from the table, tan(pi/6) = sqrt(3)/3.<BR>\n" );
document.write( "you get to the same place, but it's a little harder to figure it out, which is why i recommend converting to degrees first.\r<BR>\n" );
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document.write( "let me know if you have any questions.<BR>\n" );
document.write( "theo<BR>\n" );
document.write( " </SPAN>\n" );
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