Question 1204641
<font color=black size=3>
Law of Sines
sin(alpha)/a = sin(beta)/b
sin(alpha)/7 = sin(24)/3
sin(alpha) = 7*sin(24)/3
sin(alpha) = 0.949052
alpha = arcsin(0.949052) or alpha = 180-arcsin(0.949052)
alpha = 71.631972 or alpha = 108.368028
alpha = 71.6 or alpha = 108.4


If alpha = 71.6, then
alpha + beta + gamma = 180
71.6 + 24 + gamma = 180
gamma = 180 - 71.6 - 24
gamma = 84.4
This result is between 0 and 180, so we have a valid angle.



If alpha = 108.4, then
alpha + beta + gamma = 180
108.4 + 24 + gamma = 180
gamma = 180 - 108.4 - 24
gamma = 47.6
This result is between 0 and 180, so we have a valid angle.


We have two possible triangles.
This is the SSA (side side angle) case. Specifically it is the ambiguous case because it's not clear which triangle to pick.


Let's find the missing side c when<ul><li>alpha = 71.6 (approx)</li><li>beta = 24</li><li>gamma = 84.4 (approx)</li></ul>We can use the Law of Sines to do so
sin(gamma)/c = sin(beta)/b
sin(84.4)/c = sin(24)/3
3*sin(84.4) = c*sin(24)
c = 3*sin(84.4)/sin(24)
c = 7.340578
c = 7.3


Now let's find the missing side c when <ul><li>alpha = 108.4 (approx)</li><li>beta = 24</li><li>gamma = 47.6 (approx)</li></ul>We can use the Law of Sines
sin(gamma)/c = sin(beta)/b
sin(47.6)/c = sin(24)/3
3*sin(47.6) = c*sin(24)
c = 3*sin(47.6)/sin(24)
c = 5.44668
c = 5.4


To summarize we have two possible triangles
<table border = "1" cellpadding = "5"><tr><td></td><td>Triangle 1</td><td>Triangle 2</td></tr><tr><td>Angles</td><td>alpha = 71.6 (approx)
beta = 24
gamma = 84.4 (approx)</td><td>alpha = 108.4 (approx)
beta = 24
gamma = 47.6 (approx)</td></tr><tr><td>Sides</td><td>a = 7
b = 3
c = 7.3 (approx)</td><td>a = 7
b = 3
c = 5.4 (approx)</td></tr></table>

{{{
drawing(700,300,-10,10,-3,7,

circle(-1.29,3.99,0.05),circle(-1.29,3.99,0.07),circle(-1.29,3.99,0.09),circle(-1.29,3.99,0.11),circle(-1.29,3.99,0.13),circle(-1.29,3.99,0.15),circle(-1.29,3.99,0.17),circle(-1.29,3.99,0.19),circle(-8,1,0.05),circle(-8,1,0.07),circle(-8,1,0.09),circle(-8,1,0.11),circle(-8,1,0.13),circle(-8,1,0.15),circle(-8,1,0.17),circle(-8,1,0.19),circle(-1,1,0.05),circle(-1,1,0.07),circle(-1,1,0.09),circle(-1,1,0.11),circle(-1,1,0.13),circle(-1,1,0.15),circle(-1,1,0.17),circle(-1,1,0.19),circle(5.98,3.22,0.05),circle(5.98,3.22,0.07),circle(5.98,3.22,0.09),circle(5.98,3.22,0.11),circle(5.98,3.22,0.13),circle(5.98,3.22,0.15),circle(5.98,3.22,0.17),circle(5.98,3.22,0.19),circle(1,1,0.05),circle(1,1,0.07),circle(1,1,0.09),circle(1,1,0.11),circle(1,1,0.13),circle(1,1,0.15),circle(1,1,0.17),circle(1,1,0.19),circle(8,1,0.05),circle(8,1,0.07),circle(8,1,0.09),circle(8,1,0.11),circle(8,1,0.13),circle(8,1,0.15),circle(8,1,0.17),circle(8,1,0.19),

line(-1.29,3.99,-8,1),line(-8,1,-1,1),line(-1,1,-1.29,3.99),
line(5.98,3.22,1,1),line(1,1,8,1),line(8,1,5.98,3.22),

locate(-1.26,4.19+0.5,"A"),locate(-8.44,1.16,"B"),locate(-1,1-0.2,"C"),locate(-4.72,0.98-0.2,"a=7"),locate(-1.22+0.5,2.65,"b=3"),locate(-5.26,2.66+0.8,"c=7.3"),locate(-2.26-0.2,3.15+0.5,71.6^o),locate(-5.95,1.58+0.35,24^o),locate(-1.68-0.7,1.46+0.5,84.4^o),

locate(5.85,4.14,"A"),locate(0.56,1.13,"B"),locate(8.1,0.82,"C"),locate(4.19,0.86,"a=7"),locate(7.28,2.47,"b=3"),locate(2.81,2.67+0.2,"c=5.4"),locate(5.09,2.71,108.4^o),locate(3.01,1.86,24^o),locate(6.31,1.72+0.1,47.6^o),

locate(-6,6,matrix(1,2,"Triangle",1)),
locate(-6+10,6,matrix(1,2,"Triangle",2)),

locate(-9,-2+0.3,matrix(1,4,"Diagrams","are","to","scale")),
locate(-9,-2.5+0.3,matrix(1,4,"Decimal","values","are","approximate"))
)
}}}


I used GeoGebra to confirm each answer.


More practice with solving triangles
<a href="https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204423.html">https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204423.html</a>
</font>