Question 1151811: A= 25 degrees
a=5
b=11
Solve the triangle ABC
Thank you!!
Found 2 solutions by ankor@dixie-net.com, jim_thompson5910:
Answer by ankor@dixie-net.com(22740)   (Show Source):
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
Angles
A = 25
B = unknown
C = unknown
Sides
a = 5
b = 11
c = unknown
We have two known sides an angle that is not between the two sides. This is the SSA case which has 3 possibilities- Zero triangles are possible
- Exactly one triangle is possible
- Exactly two triangles are possible
The SSA case leads to ambiguity.
Use the law of sines to solve for angle B
sin(A)/a = sin(B)/b
sin(25)/5 = sin(B)/11
0.0845236523481399 = sin(B)/11
11*0.0845236523481399 = 11*sin(B)/11
0.929760175829539 = sin(B)
sin(B) = 0.929760175829539
arcsin(sin(B)) = arcsin(0.929760175829539)
B = arcsin(0.929760175829539) or B = 180 - arcsin(0.929760175829539)
B = 68.3974616057519 or B = 111.602538394248
B = 68.397 or B = 111.603
If B = 68.397, then C = 180-A-B = 180-25-68.397 = 86.603
If B = 111.603, then C = 180-A-B = 180-25-111.603 = 43.397
In either case, the value of B produces a positive C value. Therefore, we have two sets of solutions to represent the two possible triangles.
With this info: C = 86.603, a = 5, b = 11, we can find the unknown side c using the law of cosines
c^2 = a^2 + b^2 - 2*a*b*cos(C)
c^2 = (5)^2 + (11)^2 - 2*(5)*(11)*cos(86.603)
c^2 = 25 + 121 - 110*0.0592541057795245
c^2 = 25 + 121 - 6.51795163574769
c^2 = 139.482048364252
c = sqrt(139.482048364252)
c = 11.8102518332274
c = 11.810
So one solution consists of
Angles
A = 25
B = 68.397
C = 86.603
Sides
a = 5
b = 11
c = 11.810
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Use the law of cosines again, but now with C = 43.397
The values of 'a' and b stay the same
c^2 = a^2 + b^2 - 2*a*b*cos(C)
c^2 = (5)^2 + (11)^2 - 2*(5)*(11)*cos(43.397)
c^2 = 25 + 121 - 110*0.726610645792925
c^2 = 25 + 121 - 79.9271710372218
c^2 = 66.0728289627782
c = sqrt(66.0728289627782)
c = 8.12851948160169
c = 8.129
The other solution consists of
Angles
A = 25
B = 111.603
C = 43.397
Sides
a = 5
b = 11
c = 8.129
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In summary, the two possible triangles are
| Triangle #1 | Triangle #2 |
Angles
A = 25
B = 68.397
C = 86.603 | Sides
a = 5
b = 11
c = 11.810 | Angles
A = 25
B = 111.603
C = 43.397 | Sides
a = 5
b = 11
c = 8.129 |
Diagram
The decimal values are approximate to three decimal places.
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