How do you solve a triangle given angle A = 18.27°, angle C = 134.13°, side A = 8.3?

1 Answer
Shantelle
May 27, 2018

#color(red)(B = 27.6)#

#color(red)(c ~~ 19.003)# (rounded to nearest thousandth's place)

#color(red)(b ~~ 12.266)# (rounded to nearest thousandth's place)

Explanation:

First, we know that the angles of a triangle add up to #180^@#. We have two angles (#A# and #C#), so we can find B by doing:
#180^@ - 18.27^@ - 134.13^@ = 27.6#

Therefore, angle #color(red)(B = 27.6)#.

Now, use the Law of Sines to solve for side #c#:

#c/sinC = a/sinA#

#c/sin134.13^@ = 8.3/sin18.27^@#

#c = (8.3sin134.13)/sin18.27^@#

Therefore,
side #color(red)(c ~~ 19.003)#

Since we have two sides and an angle (or more), we can use the Law of Cosines to solve for the last side, #b#.

The Law of Cosines is #b = sqrt(a^2 + c^2 - 2(a)(c)(cosB)#

Let's plug in our values into the formula:
#b = sqrt((8.3)^2 + (19.003)^2 - 2(8.3)(19.003)(cos27.6^@))#

#b = sqrt(68.89 + 361.114 - 279.553)#

#b = sqrt(430.004 - 279.553)#

#b = sqrt(150.451)#

Therefore,
#color(red)(b ~~ 12.266)# (rounded to nearest thousandth's place)

Hope this helps!

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