document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189764>Question 1189764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> To solve triangle ABC(with angles A,B and C, sides a, b, and c, with side length la, lb, lc respectively)<BR>\n" );
document.write( "(A) If A = 45°, la=7sqrt2, and lb=7 then there is one solution.<BR>\n" );
document.write( "(B) If A=30°, la < Ib/2, then there is no solution.<BR>\n" );
document.write( "(C) If A = 30°, la > Ib/2, then there are two solutions.<BR>\n" );
document.write( "(D) All the above (A) (B) (C) are true.<BR>\n" );
document.write( "(E) None of the above (A) (B) (C) is true.<BR>\n" );
document.write( "Answer: D<BR>\n" );
document.write( "Hey, I need help to solve this please, without a calculator, i solved the first and second one(by picking random values) but i cant solve the third one. How can i do it? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821220 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=821220\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Just for clarification, something like lb is Lb with lowercase L. I'll use the uppercase version.\r<BR>\n" );
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document.write( "Angles: A, B, C<BR>\n" );
document.write( "sides: a,b,c which are opposite A,B,C in that order<BR>\n" );
document.write( "La = length of side a<BR>\n" );
document.write( "Lb = length of side b<BR>\n" );
document.write( "Lc = length of side c\r<BR>\n" );
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document.write( "Let's see if claim (A) is true<BR>\n" );
document.write( "A = 45<BR>\n" );
document.write( "La = 7*sqrt(2)<BR>\n" );
document.write( "Lb = 7\r<BR>\n" );
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document.write( "Use the law of sines<BR>\n" );
document.write( "sin(A)/La = sin(B)/Lb<BR>\n" );
document.write( "Lb*sin(A) = La*sin(B)<BR>\n" );
document.write( "7*sin(45) = 7*sqrt(2)*sin(B)<BR>\n" );
document.write( "7*sqrt(2)/2 = 7*sqrt(2)*sin(B)<BR>\n" );
document.write( "7/2 = 7sin(B) ................ the sqrt(2) terms cancel<BR>\n" );
document.write( "sin(B) = (7/2)*(1/7)<BR>\n" );
document.write( "sin(B) = 1/2<BR>\n" );
document.write( "B = arcsin(1/2)<BR>\n" );
document.write( "B = 30 or B = 180-30 = 150\r<BR>\n" );
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document.write( "If B = 30, then<BR>\n" );
document.write( "C = 180-A-B<BR>\n" );
document.write( "C = 180-45-30<BR>\n" );
document.write( "C = 105<BR>\n" );
document.write( "Showing that B = 30 is valid\r<BR>\n" );
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document.write( "If B = 150, then,<BR>\n" );
document.write( "C = 180-A-B<BR>\n" );
document.write( "C = 180-45-150<BR>\n" );
document.write( "C = -15<BR>\n" );
document.write( "Showing that B = 150 isn't possible.\r<BR>\n" );
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document.write( "Therefore, the only practical solution is B = 30<BR>\n" );
document.write( "In other words, only one triangle is possible.\r<BR>\n" );
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document.write( "We've proven claim (A) to be true. This rules out claim (E)<BR>\n" );
document.write( "Either the final answer is (A) or it's (D).\r<BR>\n" );
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document.write( "Let's check the next claim.\r<BR>\n" );
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document.write( "Claim (B) If A = 30 and La < Lb/2, then there is no solution\r<BR>\n" );
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document.write( "Let's say<BR>\n" );
document.write( "La = 10<BR>\n" );
document.write( "Lb = 30<BR>\n" );
document.write( "There's nothing special about those values. I just picked two random values to make La < Lb/2 true.\r<BR>\n" );
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document.write( "Apply the law of sines to find B<BR>\n" );
document.write( "sin(A)/La = sin(B)/Lb<BR>\n" );
document.write( "sin(45)/10 = sin(B)/30<BR>\n" );
document.write( "30*sin(45) = 10*sin(B)<BR>\n" );
document.write( "30*sqrt(2)/2 = 10*sin(B)<BR>\n" );
document.write( "15sqrt(2) = 10*sin(B)<BR>\n" );
document.write( "sin(B) = 15*sqrt(2)/10<BR>\n" );
document.write( "sin(B) = 1.5*sqrt(2)<BR>\n" );
document.write( "sin(B) = 2.1213 approximately<BR>\n" );
document.write( "We run into a problem because the largest sin(B) can get is 1. \r<BR>\n" );
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document.write( "There are no solutions to sin(B) = 2.1213\r<BR>\n" );
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document.write( "Therefore, no such triangle is possible with the properties that<BR>\n" );
document.write( "A = 30<BR>\n" );
document.write( "La = 10<BR>\n" );
document.write( "Lb = 30\r<BR>\n" );
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document.write( "More generally, no such triangle can be formed where<BR>\n" );
document.write( "A = 30<BR>\n" );
document.write( "La < Lb/2\r<BR>\n" );
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document.write( "The conclusion here is that claim (B) is true. \r<BR>\n" );
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document.write( "Claims (A) and (B) being true immediately points to choice (D) being the final answer (aka all of the above).\r<BR>\n" );
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document.write( "We don't have to keep going at this point, but for the sake of completeness, let's check claim (C)\r<BR>\n" );
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document.write( "Claim: If A = 30 and La > Lb/2, then there are two solutions\r<BR>\n" );
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document.write( "Let's say<BR>\n" );
document.write( "La = 30<BR>\n" );
document.write( "Lb = 10\r<BR>\n" );
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document.write( "sin(A)/La = sin(B)/Lb<BR>\n" );
document.write( "sin(30)/30 = sin(B)/10<BR>\n" );
document.write( "10*sin(30) = 30*sin(B)<BR>\n" );
document.write( "10*(1/2) = 30*sin(B)<BR>\n" );
document.write( "5 = 30*sin(B)<BR>\n" );
document.write( "sin(B) = 5/30<BR>\n" );
document.write( "sin(B) = 1/6<BR>\n" );
document.write( "The sin(B) being a value smaller than 1 indicates sin(B) = 1/6 is possible to solve.\r<BR>\n" );
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document.write( "Use the inverse sine function to find that the two possible B values are<BR>\n" );
document.write( "B = 9.594 or B = 180-9.594 = 170.406\r<BR>\n" );
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document.write( "If B = 9.594, then,<BR>\n" );
document.write( "C = 180-A-B<BR>\n" );
document.write( "C = 180-45-9.594<BR>\n" );
document.write( "C = 125.406<BR>\n" );
document.write( "Showing that we have a valid angle B here\r<BR>\n" );
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document.write( "Or if B = 170.406, then,<BR>\n" );
document.write( "C = 180-A-B<BR>\n" );
document.write( "C = 180-45-170.406<BR>\n" );
document.write( "C = -35.406<BR>\n" );
document.write( "That's not good. We should have gotten a positive C value.\r<BR>\n" );
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document.write( "It turns out that La > Lb/2 isn't quite the full story.\r<BR>\n" );
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document.write( "Instead, it should be<BR>\n" );
document.write( "Lb/2 < La < Lb\r<BR>\n" );
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document.write( "If that compound inequality is true and A = 30, then two triangles are possible.<BR>\n" );
document.write( "On the other hand, if La > Lb, then no triangles are possible. <BR>\n" );
document.write( "I  recommend using a tool like GeoGebra to play around with possible constructions. \r<BR>\n" );
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document.write( "It's not clear if your teacher intends La < Lb or not. It appears your teacher is implying this (or else choice D isn't possible). I would ask for further clarification.\r<BR>\n" );
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document.write( "For more info, check out this page<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Elementary_Trigonometry_(Beveridge)/04%3A_The_Law_of_Sines_and_The_Law_of_Cosines/4.02%3A_The_Law_of_Sines_-_the_ambiguous_case\">https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Elementary_Trigonometry_(Beveridge)/04%3A_The_Law_of_Sines_and_The_Law_of_Cosines/4.02%3A_The_Law_of_Sines_-_the_ambiguous_case</a>\r<BR>\n" );
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document.write( "To wrap things up, statements (A), (B), and (C) are true. This is why choice (D) is the final answer. Again this is assuming your teacher implies that La < Lb.<BR>\n" );
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