Question 1087786
<font color="black" face="times" size="3">Let,
angle A = 40 degrees
side a = 15 cm (side a is opposite angle A)
side b = 18 cm (adjacent to side a)


The ultimate goal is to find the length of side c.


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Let's use the law of sines to find angle B


sin(A)/a = sin(B)/b
sin(40)/15 = sin(B)/18
0.042852507312436 = sin(B)/18
18*0.042852507312436 = 18*sin(B)/18
0.771345131623847 = sin(B)
sin(B) = 0.771345131623847
arcsin(sin(B)) = arcsin(0.771345131623847)
B = arcsin(0.771345131623847) ... or ... B = 180 - arcsin(0.771345131623847)
B = 50.4748346316044 ... or ... B = 129.525165368396
B = 50.47 ... or ... B = 129.53
I'm rounding to two decimal places to make future calculations a bit simpler (and not as cluttered)


If angle B = 50.47, then angle C = 180-A-B = 180-40-50.47 = 89.53
If angle B = 129.53, then angle C = 180-A-B = 180-40-129.53 = 10.47


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In summary so far, we have two sets of angles


The first set of angles is
angle A = 40 degrees, angle B = 50.47 degrees (approximate), angle C = 89.53 degrees (approximate)


The other set of angles is 
angle A = 40 degrees, angle B = 129.53 degrees (approximate), angle C = 10.47 degrees (approximate)


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Let's focus on the first set of angles
angle A = 40
angle B = 50.47
angle C = 89.53
we can use these angles, along with the sides a = 15 and b = 18, to find the third side c


Use the law of cosines to find the value for c
c^2 = a^2 + b^2 - 2*a*b*cos(C)
c^2 = (15)^2 + (18)^2 - 2*(15)*(18)*cos(89.53)
c^2 = 544.570404036738
c = sqrt(544.570404036738)
c = 23.3360323113579
c = 23.34


So one possible length for the missing side is approximately 23.34 cm

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Now focus on the other set of angles 
angle A = 40
angle B = 129.53 
angle C = 10.47
we'll use these angles along with a = 15 and b = 18. Solve for c


c^2 = a^2 + b^2 - 2*a*b*cos(C)
c^2 = (15)^2 + (18)^2 - 2*(15)*(18)*cos(10.47)
c^2 = 17.9908968194146
c = sqrt(17.9908968194146)
c = 4.24156773132466
c = 4.24


The other possible length for the missing side is 4.24 cm


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The third missing side is either approximately equal to 23.34 cm or 4.24 cm


Keep in mind that I rounded things to 2 decimal places to keep things simple. Use more decimal digits to get better accuracy.


Here is a look at the two solutions side by side


<img src = "https://i.imgur.com/zGHpCW9.png">
(Image generated by <a href = "https://www.geogebra.org/home?ggbLang=en">GeoGebra</a> which is free graphing software)

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