Question 1131379
<pre>
This is the ambiguous case SSA where there are 2 solutions.
The other tutor only gave one of the solutions.  Here
are both solutions:


{{{matrix(1,5,

a/sin(A),
""="",
b/sin(B),
""="",
c/sin(C) )}}}

Substitute what's given:

{{{matrix(1,5,

a/sin(A),
""="",
51/sin(B),
""="",
50/sin("72°") )}}}

Take the last two expressions as a equation

{{{matrix(1,3,


51/sin(B),
""="",
50/sin("72°") )}}}

Cross-multiply

{{{matrix(1,3,

51sin("72°"),
""="",
50sin(B) )}}}

Divide both sides by 50

{{{matrix(1,3,

(51sin("72°"))/50,
""="",
50sin(B)/50 )}}}

{{{matrix(1,3,

51sin("72°"),
""="",
sin(B) )}}}

{{{matrix(1,3,

0.9700776466,
""="",
sin(B) )}}}

The sine is positive in QI and QII, so

B = 75.94844394° and B = 180°-75.94844394 = 104.0515561 

Let's subscript the two possibilities for angle B like this:

B<sub>1</sub> = 75.94844394° and B<sub>2</sub> = 104.0515561° 

Now we calculate the two possibilities for angle A:

Angle A<sub>1</sub> = 180° - Angle B<sub>1</sub> - Angle C
Angle A<sub>1</sub> = 180° - 75.94844394° - 72°
Angle A<sub>1</sub> = 32.05155605°

Angle A<sub>2</sub> = 180° - Angle B<sub>2</sub> - Angle C
Angle A<sub>2</sub> = 180° - 104.0515561°  - 72°
Angle A<sub>2</sub> = 3.9484439°

Finally we calculate the two possibilities for side "a":
----------

{{{matrix(1,5,

a[1]/sin(A[1]),
""="",
b[""]/sin(B[1]),
""="",
c[""]/sin(C) )}}} and {{{matrix(1,5,

a[2]/sin(A[2]),
""="",
b[""]/sin(B[2]),
""="",
c[""]/sin(C) )}}} 

Substitute what is now known:

{{{matrix(1,5,

a[1]/sin("32.0515561°"[""]),
""="",
51[""]/sin("75.94844394°"[""]),
""="",
50[""]/sin("72°"[""]) )}}} and {{{matrix(1,5,

a[2]/sin("3.9484439°"[""]),
""="",
51[""]/sin("104.0515561°"[""]),
""="",
50[""]/sin("72°"[""]) )}}} 

which shortens to:

{{{matrix(1,3,

a[1]/sin("32.0515561°"[""]),
""="",
50[""]/sin("72°"[""]) )}}} and {{{matrix(1,3,

a[2]/sin("3.9484439°"[""]),
""="",
50[""]/sin("72°"[""]) )}}}

{{{matrix(1,3,

a[1]sin("72°"[""]),
""="",
50[""]sin("32.0515561°"[""]) )}}} and {{{matrix(1,3,

a[2]sin("72°"[""]),
""="",
50[""]sin("3.9484439°"[""]) )}}}

Divide through by sin(72°)

{{{matrix(1,3,

a[1],
""="",
50[""]sin("32.0515561°"[""])/sin("72°"[""]) )}}} and {{{matrix(1,3,

a[2],
""="",
50[""]sin("3.9484439°"[""])/sin("72°"[""]) )}}}

{{{matrix(1,3,

a[1],
""="",
27.89961134 )}}} and {{{matrix(1,3,

a[2],
""="",
3.620122079 )}}}

Edwin</pre>