Respuesta :
Answer:
[tex]\angle B_{1} \approx 31.668^{\circ}[/tex], [tex]\angle B_{2} \approx 148.332^{\circ}[/tex]
[tex]\angle C_{1} \approx 127.332^{\circ}[/tex], [tex]\angle C_{2} \approx 10.668^{\circ}[/tex]
[tex]c_{1} \approx 157.532[/tex], [tex]c_{2}\approx 36.676[/tex]
Step-by-step explanation:
The Law of Sines states that:
[tex]\frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
Where:
[tex]a[/tex], [tex]b[/tex], [tex]c[/tex] - Side lengths, dimensionless.
[tex]A[/tex], [tex]B[/tex], [tex]C[/tex] - Angles opposite to respective sides, dimensionless.
Given that [tex]a = 71[/tex], [tex]b = 104[/tex], [tex]\angle A = 21^{\circ}[/tex], the sine of angle B is:
[tex]\sin B = \frac{b}{a}\cdot \sin A[/tex]
[tex]\sin B = \frac{104}{71}\cdot \sin 21^{\circ}[/tex]
[tex]\sin B = 0.525[/tex]
Sine is positive between 0º and 180º, so there are two possible solutions:
[tex]\angle B_{1} \approx 31.668^{\circ}[/tex]
[tex]\angle B_{2} \approx 148.332^{\circ}[/tex]
The remaining angle is obtained from the principle that sum of internal triangles equals to 180 degrees: ([tex]\angle A = 21^{\circ}[/tex], [tex]\angle B_{1} \approx 31.668^{\circ}[/tex], [tex]\angle B_{2} \approx 148.332^{\circ}[/tex])
[tex]\angle C_{1} = 180^{\circ}-\angle A - \angle B_{1}[/tex]
[tex]\angle C_{1} = 180^{\circ}-21^{\circ}-31.668^{\circ}[/tex]
[tex]\angle C_{1} \approx 127.332^{\circ}[/tex]
[tex]\angle C_{2} = 180^{\circ}-\angle A - \angle B_{2}[/tex]
[tex]\angle C_{2} = 180^{\circ}-21^{\circ}-148.332^{\circ}[/tex]
[tex]\angle C_{2} \approx 10.668^{\circ}[/tex]
Lastly, the remaining side of the triangle is found by means of the Law of Sine: ([tex]a = 71[/tex], [tex]\angle A = 21^{\circ}[/tex], [tex]\angle C_{1} \approx 127.332^{\circ}[/tex], [tex]\angle C_{2} \approx 10.668^{\circ}[/tex])
[tex]c_{1} = a\cdot \left(\frac{\sin C_{1}}{\sin A} \right)[/tex]
[tex]c_{1} = 71\cdot \left(\frac{\sin 127.332^{\circ}}{\sin 21^{\circ}} \right)[/tex]
[tex]c_{1} \approx 157.532[/tex]
[tex]c_{2} = a\cdot \left(\frac{\sin C_{2}}{\sin A} \right)[/tex]
[tex]c_{2}= 71\cdot \left(\frac{\sin 10.668^{\circ}}{\sin 21^{\circ}} \right)[/tex]
[tex]c_{2}\approx 36.676[/tex]
The answer are presented below:
[tex]\angle B_{1} \approx 31.668^{\circ}[/tex], [tex]\angle B_{2} \approx 148.332^{\circ}[/tex]
[tex]\angle C_{1} \approx 127.332^{\circ}[/tex], [tex]\angle C_{2} \approx 10.668^{\circ}[/tex]
[tex]c_{1} \approx 157.532[/tex], [tex]c_{2}\approx 36.676[/tex]