document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206406>Question 1206406</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with this equation.<BR>\n" );
document.write( "Find sin(2x), cos(2x), and tan(2x) from the given information.<BR>\n" );
document.write( "tan(x) = −4/3, x in Quadrant II<BR>\n" );
document.write( "sin(2x)	=	<BR>\n" );
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document.write( "cos(2x)	=	<BR>\n" );
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document.write( "tan(2x)	=	<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843844 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll replace x with the Greek letter theta (symbol <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >) since it is more commonly used with trig angles. <BR>\n" );
document.write( "Also it's to avoid potential upcoming confusion when I'll use x in a different way.\r<BR>\n" );
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document.write( "Given: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29+=+-4%2F3 ALIGN=MIDDLE  ALT=\"tan%28theta%29+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > where angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in quadrant II.\r<BR>\n" );
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document.write( "Goal: Compute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29 ALIGN=MIDDLE  ALT=\"sin%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29 ALIGN=MIDDLE  ALT=\"cos%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Quadrant II is the northwest quadrant i.e. upper left corner.<BR>\n" );
document.write( "This is for any point (x,y) such that x < 0 and y > 0.\r<BR>\n" );
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document.write( "Here's what the diagram looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C6%2C-6%2C6%2C%0D%0Agraph%28400%2C400%2C-6%2C6%2C-6%2C6%29%2C%0D%0Aline%280%2C0%2C-3%2C0%29%2C%0D%0Aline%28-3%2C0%2C-3%2C4%29%2C%0D%0Aline%28-3%2C4%2C0%2C0%29%2C%0D%0Alocate%28-4%2C3%2C%22y%22%29%2C%0D%0Alocate%28-5.5%2C2%2C%22opposite%22%29%2C%0D%0Alocate%28-2%2C-1%2B0.3%2C%22x%22%29%2C%0D%0Alocate%28-2.7%2C-1.5%2B0.3%2C%22adjacent%22%29%2C%0D%0Alocate%28-2%2B0.7%2C3%2C%22r%22%29%2C%0D%0Alocate%28-2.7%2B0.7%2C4%2C%22hypotenuse%22%29%2C%0D%0Alocate%280.6%2C1%2Ctheta%29%2C%0D%0Aarc%280%2C0%2C1.5%2C1.5%2C233.1301023542%2C360%29%2C%0D%0Alocate%28-5.5%2C5.5%2Cmatrix%281%2C2%2C%22Quadrant%22%2C%22II%22%29%29%2C%0D%0Alocate%28-5.5%2C-5%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22is%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Tangent is the ratio of opposite over adjacent. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22tan%28angle%29%22+=+%22opposite%22%2F%22adjacent%22 ALIGN=MIDDLE  ALT=\"%22tan%28angle%29%22+=+%22opposite%22%2F%22adjacent%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the same as saying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29+=+y%2Fx ALIGN=MIDDLE  ALT=\"tan%28theta%29+=+y%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y = opposite<BR>\n" );
document.write( "x = adjacent\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29+=+-4%2F3 ALIGN=MIDDLE  ALT=\"tan%28theta%29+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > then we would have y = 4 and x = -3 so the x < 0 and y > 0 requirements are fulfilled.<BR>\n" );
document.write( "The terminal point is located at (x,y) = (-3,4).\r<BR>\n" );
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document.write( "Now let's find the hypotenuse of this right triangle using the Pythagorean theorem. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2+=+c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"r+=+sqrt%28x%5E2%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%28%28-3%29%5E2%2B%284%29%5E2%29 ALIGN=MIDDLE  ALT=\"r+=+sqrt%28%28-3%29%5E2%2B%284%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%2825%29 ALIGN=MIDDLE  ALT=\"r+=+sqrt%2825%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+5 ALIGN=MIDDLE  ALT=\"r+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The hypotenuse is 5 units long.<BR>\n" );
document.write( "We have a 3-4-5 right triangle.\r<BR>\n" );
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document.write( "y = opposite = 4<BR>\n" );
document.write( "x = adjacent = -3<BR>\n" );
document.write( "r = hypotenuse = 5\r<BR>\n" );
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document.write( "which means,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+y%2Fr ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+y%2Fr\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+4%2F5 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29+=+x%2Fr ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+x%2Fr\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29+=+-3%2F5 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+-3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Furthermore,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29+=+2%2Asin%28theta%29%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"sin%282theta%29+=+2%2Asin%28theta%29%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29+=+2%2A%284%2F5%29%2A%28-3%2F5%29 ALIGN=MIDDLE  ALT=\"sin%282theta%29+=+2%2A%284%2F5%29%2A%28-3%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29+=+-24%2F25 ALIGN=MIDDLE  ALT=\"sin%282theta%29+=+-24%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=red>which is one of the answers</font><BR>\n" );
document.write( "and also<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29+=+cos%5E2%28theta%29+-+sin%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"cos%282theta%29+=+cos%5E2%28theta%29+-+sin%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > see <font color=green>note</font> below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29+=+%28-3%2F5%29%5E2+-+%284%2F5%29%5E2 ALIGN=MIDDLE  ALT=\"cos%282theta%29+=+%28-3%2F5%29%5E2+-+%284%2F5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29+=+9%2F25+-+16%2F25 ALIGN=MIDDLE  ALT=\"cos%282theta%29+=+9%2F25+-+16%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29+=+-7%2F25 ALIGN=MIDDLE  ALT=\"cos%282theta%29+=+-7%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=red>which is another answer</font><BR>\n" );
document.write( "lastly<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29+=+%28sin%282theta%29%29%2F%28cos%282theta%29%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29+=+%28sin%282theta%29%29%2F%28cos%282theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > see <font color=green>note</font> below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29+=+matrix%281%2C3%2Csin%282theta%29%2C%22divide%22%2Ccos%282theta%29%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29+=+matrix%281%2C3%2Csin%282theta%29%2C%22divide%22%2Ccos%282theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29+=+matrix%281%2C3%2C%28-24%2F25%29%2C%22divide%22%2C%28-7%2F25%29%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29+=+matrix%281%2C3%2C%28-24%2F25%29%2C%22divide%22%2C%28-7%2F25%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29+=+%28-24%2F25%29%2A%28-25%2F7%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29+=+%28-24%2F25%29%2A%28-25%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29+=+24%2F7 ALIGN=MIDDLE  ALT=\"tan%282theta%29+=+24%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=red>which is the last answer we need</font>\r<BR>\n" );
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document.write( "<font color=green>Note</font>: There are other identities you could use as an alternative pathway.<BR>\n" );
document.write( "Refer to this list of trig identities<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf\">https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf</a><BR>\n" );
document.write( "Specifically refer to the \"double angle formulas\" section on page 2.\r<BR>\n" );
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document.write( "A summary of the answers:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29+=+-24%2F25 ALIGN=MIDDLE  ALT=\"sin%282theta%29+=+-24%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29+=+-7%2F25 ALIGN=MIDDLE  ALT=\"cos%282theta%29+=+-7%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29+=+24%2F7 ALIGN=MIDDLE  ALT=\"tan%282theta%29+=+24%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "</font> </SPAN>\n" );
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