document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=849814>Question 849814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the value of sin(A+B) if 5cos A-4=0 and 12tan B+5=0? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #511764 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=511764\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 5cosA-4 = 0<BR>\n" );
document.write( "add 4 to both sides of this equation to get:<BR>\n" );
document.write( "5cosA = 4<BR>\n" );
document.write( "divide both sides of this equation by 5 to get:<BR>\n" );
document.write( "cosA = 4/5\r<BR>\n" );
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document.write( "12tanB+5 = 0<BR>\n" );
document.write( "subtract 5 from  both sides of this equation to get:<BR>\n" );
document.write( "12tanB = -5<BR>\n" );
document.write( "divide both sides of this equation by 12 to get:<BR>\n" );
document.write( "tanB = -5/12\r<BR>\n" );
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document.write( "when the cosine of angle A is positive, angle A can be in either the first quadrant or the fourth quadrant.\r<BR>\n" );
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document.write( "when the tangent of angle B is negative, angle B can be in either the second quadrant or the fourth quadrant.\r<BR>\n" );
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document.write( "i usually find it easiest to find the angle in the first quadrant and then do the transposition from there.\r<BR>\n" );
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document.write( "if the angle is in the first quadrant then the cosine and the tangent will both be positive.\r<BR>\n" );
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document.write( "angle A in the first quadrant is equal to arccos (4/5) which is equal to 36.86989765 degrees.\r<BR>\n" );
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document.write( "angle B in the first quadrant is equal to arctan (5/12) which is equal to 22.61986495.\r<BR>\n" );
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document.write( "now that the angle has been found in the first quadrant, it is relatively simple to find the angle in other quadrants.\r<BR>\n" );
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document.write( "The cosine of an angle is positive when the angle is in the first quadrant or the fourth quadrant\r<BR>\n" );
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document.write( "in the first quadrant, angle A is equal to 36.86989765 degrees.<BR>\n" );
document.write( "in the fourth quadrant, angle A is equal to 360 - 36.86989765 degrees which is equal to 323.1301025 degrees.<BR>\n" );
document.write( "to verify this is accurate, use your calculator to find the cosine of 36.86989765 degrees and the cosine of 323.1301025 degrees.<BR>\n" );
document.write( "you should find that both cosines are positive and are equal to 4/5.\r<BR>\n" );
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document.write( "the tangent of an angle is negative when the angle is in the second quadrant and the fourth quadrant.<BR>\n" );
document.write( "in the second quadrant, angle B is equal to 180 - 22.61986495 degrees which is equal to 157.3801351 degrees.<BR>\n" );
document.write( "in the fourth quadrant, angle B is equal to 360 - 22.61986495 degrees which is equal to 337.3801351 degrees.<BR>\n" );
document.write( "to verify this is accurate, use your calculator to find the tangent of 157.3801351 degrees and the tangent of 337.3801351 degrees.<BR>\n" );
document.write( "you should find that both tangents are negative and are equal to 5/12.\r<BR>\n" );
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document.write( "i checked with my calculator and it confirms that these values are good.\r<BR>\n" );
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document.write( "you are not done yet though.\r<BR>\n" );
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document.write( "you want to find sin(A+B).\r<BR>\n" );
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document.write( "you need to add the angles together and then find the sine.\r<BR>\n" );
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document.write( "since A is in the first quadrant and the fourth quadrant, and B is in the second quadrant and the fourth quadrant, you have 2 * 2 = 4 possible combinations.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "A in the first quadrant plus B in the second quadrant.<BR>\n" );
document.write( "A in the first quadrant plus B in the fourth quadrant.<BR>\n" );
document.write( "A in the fourth quadrant plus B in the second quadrant.<BR>\n" );
document.write( "A in the fourth quadrant plus B in the fourth quadrant.\r<BR>\n" );
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document.write( "here's where it gets a little murky for me.\r<BR>\n" );
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document.write( "i believe you want to find the quadrant that both angles are in.<BR>\n" );
document.write( "this allows you to get one solution.\r<BR>\n" );
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document.write( "both angles are in the fourth quadrant.<BR>\n" );
document.write( "in the fourth quadrant, sine is negative and cosine is positive and tangent is negative.\r<BR>\n" );
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document.write( "the fourth quadrant satisfies the requirement that cosine (A) is positive and tangent (B) is negative.\r<BR>\n" );
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document.write( "i believe you want to find sin(A+B) in the fourth quadrant.\r<BR>\n" );
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document.write( "in the fourth quadrant,the sine of the angle is negative.<BR>\n" );
document.write( "angle A is equal to 323.1301024 degrees.<BR>\n" );
document.write( "angle B is equal to 337.3801351 degrees.<BR>\n" );
document.write( "add these 2 angles together and you get an angle of 660.5102374 degrees.<BR>\n" );
document.write( "use your calculator to find the sine of this angle and you get:<BR>\n" );
document.write( "sin(660.5102374) = -.8615384615.\r<BR>\n" );
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document.write( "if my assumption is correct, that's your answer.\r<BR>\n" );
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document.write( "you could  also have gotten the equivalent angle less than 360 degrees by subtracting 360 from 660 to get angle (A+B) = 300.5102374 degrees.<BR>\n" );
document.write( "sin(300.5102374) = -.8615384615<BR>\n" );
document.write( "this is to be expected since we are talking about the same angle.\r<BR>\n" );
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document.write( "i also looked at both angles as if they were in the first quadrant and i got an angle of 59.48976259 that produced a sine of .8615384615.\r<BR>\n" );
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document.write( "take that angle of 59.48976259 degrees and transpose it into the fourth quadrant and you get an angle of 360 - 59.48976259 degrees which is equal to 300.5102374 degrees.\r<BR>\n" );
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document.write( "an angle of 300.5102374 degrees is also equivalent to an angle of -59.48976259 degrees.\r<BR>\n" );
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document.write( "I'm pretty sure I'm correct.<BR>\n" );
document.write( "I'm very sure up to finding the sine of (A+B)<BR>\n" );
document.write( "I'm pretty sure after that.\r<BR>\n" );
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