document.write( "Question 479139: solve triangle ABC by using the Law of sine. A=115, a=70m, b=31m. Round all sides and angles to the nearest tenth.\r
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\n" ); document.write( "\n" ); document.write( "Solve triangle ABC by using the Law of Cosines. B=106, a=14.6, c=10.5. Round all sides and angles to the nearest tenth. \n" ); document.write( "
Algebra.Com's Answer #328318 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
problem number 1:
\n" ); document.write( "law of sines is:
\n" ); document.write( "a/sin(A) = b/sin(B) = c/sin(C)
\n" ); document.write( "you have:
\n" ); document.write( "a = 70m
\n" ); document.write( "b = 31m
\n" ); document.write( "A = 115 degrees.
\n" ); document.write( "the part of the law can use is:
\n" ); document.write( "a/sin(A) = b/sin(B)
\n" ); document.write( "this equation becomes:
\n" ); document.write( "70/sin(115) = 31/sin(B)
\n" ); document.write( "multiply both sides of this equation by sin(115) and multiply both sides of this equation by sin(B) and you get:
\n" ); document.write( "70 * sin(B) = 31 * sin(115)
\n" ); document.write( "this is the same thing you would do if i told you to cross multiply.
\n" ); document.write( "now divide both sides of the equation by 70 to get:
\n" ); document.write( "sin(B) = (31 * sin(115) / 70
\n" ); document.write( "use your calculator to solve for sin(B) to get:
\n" ); document.write( "sin(B) = (31 * .906307787) / 70 which becomes:
\n" ); document.write( "sin(B) = 28.0955414 / 70 which becomes:
\n" ); document.write( "sin(B) = .401364877
\n" ); document.write( "use your calculator to solve for B to get:
\n" ); document.write( "B = arc sin(.401364877) = 23.6635313 degrees.
\n" ); document.write( "confirm by plugging into your original equation to get:
\n" ); document.write( "a/sin(A) = b/sin(B) becomes:
\n" ); document.write( "31/.401364877 = 70/.906307787 which becomes:
\n" ); document.write( "77.23645433 = 77.23645433, confirming the answer for angle B is good.
\n" ); document.write( "your answer is:
\n" ); document.write( "A = 23.6635313 degrees
\n" ); document.write( "you now have:
\n" ); document.write( "a = 70m
\n" ); document.write( "b = 31m
\n" ); document.write( "A = 115 degrees.
\n" ); document.write( "B = 23.6635313 degrees.
\n" ); document.write( "C = 180 - A - B = 41.3364687
\n" ); document.write( "this is because the sum of the angles of a triangle must be equal to 180 degrees.
\n" ); document.write( "you can use the law of sines again to solve for c.
\n" ); document.write( "you can use either:
\n" ); document.write( "a/sin(A) = c/sin(C)
\n" ); document.write( "or you can use:
\n" ); document.write( "b/sin(B) = c/sin(C)
\n" ); document.write( "either one will get you the answer.
\n" ); document.write( "using b/sin(B) = c/sin(C), we get:
\n" ); document.write( "31/sin(23.6635313) = c/sin(41.3364687), we get:
\n" ); document.write( "c = (31 * sin(41.3364687))/sin(23.6635313) which gets:
\n" ); document.write( "c = 51.0131112
\n" ); document.write( "you now have:
\n" ); document.write( "a = 70m
\n" ); document.write( "b = 31m
\n" ); document.write( "c = 51.0131112
\n" ); document.write( "A = 115 degrees.
\n" ); document.write( "B = 23.6635313 degrees.
\n" ); document.write( "C = 41.3364687
\n" ); document.write( "confirm the answer is correct by taking any angle and deriving the common ratio.
\n" ); document.write( "take a/sin(A) to derive k = 77.23645433
\n" ); document.write( "if all is good, then:
\n" ); document.write( "b/k = sin(B) which makes B = 23.6635313
\n" ); document.write( "c/k = sin(C) which makes C = 41.3364687
\n" ); document.write( "all numbers check out which means that you applied the law of sines correctly.
\n" ); document.write( "-----
\n" ); document.write( "problem number 2:
\n" ); document.write( "law of cosines is:
\n" ); document.write( "law of cosines is:
\n" ); document.write( "c^2 = a^2 + b^2 - 2*a*b*cos(C)
\n" ); document.write( "it is also:
\n" ); document.write( "b^2 = a^2 + c^2 - 2*a*c*cos(B)
\n" ); document.write( "it is also:
\n" ); document.write( "a^2 = b^2 + c^2 - 2*b*c*cos(A)
\n" ); document.write( "-----
\n" ); document.write( "you are given:
\n" ); document.write( "a = 14.6
\n" ); document.write( "c = 10.5
\n" ); document.write( "B = 106 degrees
\n" ); document.write( "the law of cosines formula you would use is:
\n" ); document.write( "b^2 = a^2 + c^2 - 2*a*c*cos(B)
\n" ); document.write( "replace a and c and angle B with the value you are given to get:
\n" ); document.write( "b^2 = 14.6^2 + 10.5^2 - 2*14.6*10.5*cos(106)
\n" ); document.write( "simplify this to get:
\n" ); document.write( "b^2 = 213.16 + 110.25 - 306.6 * cos(106)
\n" ); document.write( "simplify further to get:
\n" ); document.write( "b^2 = 323.41 - 306.6*cos(106)
\n" ); document.write( "use your calculator to find cos(106) and plug it into the equation to get:
\n" ); document.write( "b^2 = 323.41 - 306.6 * (-.275637356)
\n" ); document.write( "simplify to get:
\n" ); document.write( "b^2 = 323.41 - (-84.51041329)
\n" ); document.write( "simplify further to get:
\n" ); document.write( "b^2 = 323.41 + 84.51041329
\n" ); document.write( "simplify further to get:
\n" ); document.write( "b^2 = 407.9204133
\n" ); document.write( "take the square root of both sides of the equation to get:
\n" ); document.write( "b = 20.19703972
\n" ); document.write( "it's actually +/- 20.19703972 but b can't be negative so you are left with the positive square root.
\n" ); document.write( "you now have:
\n" ); document.write( "a = 14.6
\n" ); document.write( "b = 20.19703972
\n" ); document.write( "c = 10.5
\n" ); document.write( "B = 106 degrees
\n" ); document.write( "you can use the law of cosines again to solve for angle A.
\n" ); document.write( "the law of cosines that you would use would be:
\n" ); document.write( "a^2 = b^2 + c^2 - 2*b*c*cos(A)
\n" ); document.write( "you would plug known values into that equation to get:
\n" ); document.write( "14.6^2 = 20.19703972^2 + 10.5^2 - 2 * 20.19703972 * 10.5 * cos(A)
\n" ); document.write( "simplify this to get:
\n" ); document.write( "213.16 = 407.9204133 + 110.25 - 424.1378341 * cos(A)
\n" ); document.write( "simplify further to get:
\n" ); document.write( "213.16 = 518.1704133 - 424.137834 * cos(A)
\n" ); document.write( "subtract 518.1704133 from both sides of the equation to get:
\n" ); document.write( "-305.0104133 = -424.137834 * cos(A)
\n" ); document.write( "divide both sides of the equation by -424.137834 to get:
\n" ); document.write( ".719130407 = cos(A) which gets:
\n" ); document.write( "A = 44.01726815 degrees
\n" ); document.write( "you now have:
\n" ); document.write( "a = 14.6
\n" ); document.write( "b = 20.19703972
\n" ); document.write( "c = 10.5
\n" ); document.write( "A = 44.01726815 degrees
\n" ); document.write( "B = 106 degrees
\n" ); document.write( "you can derive C by taking 180 - A - B to get:
\n" ); document.write( "C = 29.98273185
\n" ); document.write( "this triangle is complete.
\n" ); document.write( "if you did this accurately, then the law of sines should also work with this triangle.
\n" ); document.write( "b/sin(B) to get the common ratio.
\n" ); document.write( "that would be k = 21.01096903
\n" ); document.write( "a/k should be equal to sin(A) which gets A = 44.01726815 degrees
\n" ); document.write( "c/k should be equal to sin(C) which gets C = 29.98273183 degrees
\n" ); document.write( "all numbers check out which means that you applied the law of cosines correctly.\r
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