SOLUTION: solve triangle ABC by using the Law of sine. A=115, a=70m, b=31m. Round all sides and angles to the nearest tenth. Solve triangle ABC by using the Law of Cosines. B=106, a=14.

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.

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

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