Question 1087786: Hey guys I been trying to solve this question but I just don't get it
The question is;
In constructing a triangle, the direction were to make one angle 40° with its opposite side 15 cm and an adjacent side 18 cm long. What are the possible measurements for the third side?
Your help will be very much appreciated, THANK YOU
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! 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
(Image generated by GeoGebra which is free graphing software)
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