Answer:
, ![\angle B_{2} \approx 148.332^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20B_%7B2%7D%20%5Capprox%20148.332%5E%7B%5Ccirc%7D)
, ![\angle C_{2} \approx 10.668^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B2%7D%20%5Capprox%2010.668%5E%7B%5Ccirc%7D)
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Step-by-step explanation:
The Law of Sines states that:
![\frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C}](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7B%5Csin%20A%7D%20%3D%20%5Cfrac%7Bb%7D%7B%5Csin%20B%7D%3D%5Cfrac%7Bc%7D%7B%5Csin%20C%7D)
Where:
,
,
- Side lengths, dimensionless.
,
,
- Angles opposite to respective sides, dimensionless.
Given that
,
,
, the sine of angle B is:
![\sin B = \frac{104}{71}\cdot \sin 21^{\circ}](https://tex.z-dn.net/?f=%5Csin%20B%20%3D%20%5Cfrac%7B104%7D%7B71%7D%5Ccdot%20%5Csin%2021%5E%7B%5Ccirc%7D)
![\sin B = 0.525](https://tex.z-dn.net/?f=%5Csin%20B%20%3D%200.525)
Sine is positive between 0º and 180º, so there are two possible solutions:
![\angle B_{1} \approx 31.668^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20B_%7B1%7D%20%5Capprox%2031.668%5E%7B%5Ccirc%7D)
![\angle B_{2} \approx 148.332^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20B_%7B2%7D%20%5Capprox%20148.332%5E%7B%5Ccirc%7D)
The remaining angle is obtained from the principle that sum of internal triangles equals to 180 degrees: (
,
,
)
![\angle C_{1} = 180^{\circ}-\angle A - \angle B_{1}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B1%7D%20%3D%20180%5E%7B%5Ccirc%7D-%5Cangle%20A%20-%20%5Cangle%20B_%7B1%7D)
![\angle C_{1} = 180^{\circ}-21^{\circ}-31.668^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B1%7D%20%3D%20180%5E%7B%5Ccirc%7D-21%5E%7B%5Ccirc%7D-31.668%5E%7B%5Ccirc%7D)
![\angle C_{1} \approx 127.332^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B1%7D%20%5Capprox%20127.332%5E%7B%5Ccirc%7D)
![\angle C_{2} = 180^{\circ}-\angle A - \angle B_{2}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B2%7D%20%3D%20180%5E%7B%5Ccirc%7D-%5Cangle%20A%20-%20%5Cangle%20B_%7B2%7D)
![\angle C_{2} = 180^{\circ}-21^{\circ}-148.332^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B2%7D%20%3D%20180%5E%7B%5Ccirc%7D-21%5E%7B%5Ccirc%7D-148.332%5E%7B%5Ccirc%7D)
![\angle C_{2} \approx 10.668^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B2%7D%20%5Capprox%2010.668%5E%7B%5Ccirc%7D)
Lastly, the remaining side of the triangle is found by means of the Law of Sine: (
,
,
,
)
![c_{1} = a\cdot \left(\frac{\sin C_{1}}{\sin A} \right)](https://tex.z-dn.net/?f=c_%7B1%7D%20%3D%20a%5Ccdot%20%5Cleft%28%5Cfrac%7B%5Csin%20C_%7B1%7D%7D%7B%5Csin%20A%7D%20%5Cright%29)
![c_{1} = 71\cdot \left(\frac{\sin 127.332^{\circ}}{\sin 21^{\circ}} \right)](https://tex.z-dn.net/?f=c_%7B1%7D%20%3D%2071%5Ccdot%20%5Cleft%28%5Cfrac%7B%5Csin%20127.332%5E%7B%5Ccirc%7D%7D%7B%5Csin%2021%5E%7B%5Ccirc%7D%7D%20%5Cright%29)
![c_{1} \approx 157.532](https://tex.z-dn.net/?f=c_%7B1%7D%20%5Capprox%20157.532)
![c_{2} = a\cdot \left(\frac{\sin C_{2}}{\sin A} \right)](https://tex.z-dn.net/?f=c_%7B2%7D%20%3D%20a%5Ccdot%20%5Cleft%28%5Cfrac%7B%5Csin%20C_%7B2%7D%7D%7B%5Csin%20A%7D%20%5Cright%29)
![c_{2}= 71\cdot \left(\frac{\sin 10.668^{\circ}}{\sin 21^{\circ}} \right)](https://tex.z-dn.net/?f=c_%7B2%7D%3D%2071%5Ccdot%20%5Cleft%28%5Cfrac%7B%5Csin%2010.668%5E%7B%5Ccirc%7D%7D%7B%5Csin%2021%5E%7B%5Ccirc%7D%7D%20%5Cright%29)
The answer are presented below:
, ![\angle B_{2} \approx 148.332^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20B_%7B2%7D%20%5Capprox%20148.332%5E%7B%5Ccirc%7D)
, ![\angle C_{2} \approx 10.668^{\circ}](https://tex.z-dn.net/?f=%5Cangle%20C_%7B2%7D%20%5Capprox%2010.668%5E%7B%5Ccirc%7D)
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