SOLUTION: Solve the remaining sides and angles of the Triangle A=36degrees C=105degrees c=52 solve a, b, B,

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Question 1131378: Solve the remaining sides and angles of the Triangle
A=36degrees
C=105degrees
c=52
solve a, b, B,
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:
A=36°
C=105°
c=52

solution:
B=180-%28A%2BC%29
B=180-%2836%2B105%29
B=180-141
highlight%28B=39%29°


use law of cos
a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29
b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29
=>
a=sqrt%28b%5E2%2Bc%5E2-2bc%2Acos%28A%29%29
b=sqrt%28a%5E2%2Bc%5E2-2ac%2Acos%28B%29%29

if given c=52, A=36°, B=39°, we have


a%5E2+=+b%5E2+%2B+52%5E2+-2b%2A52+cos%2836%29
a%5E2+=+b%5E2+%2B+%28-26+-+26+sqrt%285%29%29+b+%2B+2704

a%5E2+=+b%5E2+-+84.14b+%2B+2704+..............eq.1

=>a+=sqrt%28+b%5E2+-+84.14b+%2B+2704%29+....eq.1a

now,
b%5E2=a%5E2%2B52%5E2-2a%2A52%2Acos%28B%29
b%5E2+=+a%5E2++-+80.8a+%2B+2704....substitute a%5E2-eq.1 and a-eq.1a
b%5E2+=+b%5E2+-+84.14b+%2B+2704+%2B+2704+-80.8sqrt%28+b%5E2+-+84.14b+%2B+2704%29+
b%5E2+-+b%5E2+=-+84.14b++-80.8sqrt%28+b%5E2+-+84.14b+%2B+2704%29+%2B5408
0+=-+84.14b++-80.8sqrt%28+b%5E2+-+84.14b+%2B+2704%29+%2B5408
+84.14b-+5408+=-80.8sqrt%28+b%5E2+-+84.14b+%2B+2704%29+...square both sides
%28+84.14b-+5408%29%5E2+=%28-80.8sqrt%28+b%5E2+-+84.14b+%2B+2704%29+%29%5E2
7079.54b%5E2+-+910058%2Ab+%2B+29246464=6528.64%28+b%5E2+-+84.14b+%2B+2704%29+
7079.54b%5E2+-+910058%2Ab+%2B+29246464=6528.64+b%5E2+-+549320.+b+%2B+17653442.56+
7079.54b%5E2+-+910058%2Ab+%2B+29246464-6528.64+b%5E2+%2B+549320.+b+-17653442.56+=0
550.9b%5E2+-+360738%2Ab+%2B+11593000=0
550.9%28b+-+620.925%29%28b+-+33.89%29+=+0
highlight%28b=33.89%29

substitute in eq.1a
a+=sqrt%28+b%5E2+-+84.14b+%2B+2704%29+....eq.1a
a+=sqrt%28+%2833.89%29%5E2+-+84.14%2A33.89+%2B+2704%29+
a+=sqrt%28+1148.5321+-+2851.5046%2B+2704%29+
a+=sqrt%281001.0275%29+
highlight%28a+=31.64%29+


Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the remaining sides and angles of the Triangle
A=36degrees
C=105degrees
c=52
solve a, b, B, 
∠B = 180o-(36o + 105o)

To find "a," use LAW of SINES. Use: 

To find "b," use LAW of SINES. Use: