Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. Below, enter your answers so that ∠A1 is smaller than ∠A2.)
a = 71, b = 104, ∠A = 21°
∠B1 = ∠B2 =
∠C1 = ∠C2 =
c1 = c2 =

Answer 1

Answer:

$\angle B_{1} \approx 31.668^{\circ}$, $\angle B_{2} \approx 148.332^{\circ}$

$\angle C_{1} \approx 127.332^{\circ}$, $\angle C_{2} \approx 10.668^{\circ}$

$c_{1} \approx 157.532$, $c_{2}\approx 36.676$

Step-by-step explanation:

The Law of Sines states that:

$\frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C}$

Where:

$a$, $b$, $c$ - Side lengths, dimensionless.

$A$, $B$, $C$ - Angles opposite to respective sides, dimensionless.

Given that $a = 71$, $b = 104$, $\angle A = 21^{\circ}$, the sine of angle B is:

$\sin B = \frac{b}{a}\cdot \sin A$

$\sin B = \frac{104}{71}\cdot \sin 21^{\circ}$

$\sin B = 0.525$

Sine is positive between 0º and 180º, so there are two possible solutions:

$\angle B_{1} \approx 31.668^{\circ}$

$\angle B_{2} \approx 148.332^{\circ}$

The remaining angle is obtained from the principle that sum of internal triangles equals to 180 degrees: ($\angle A = 21^{\circ}$, $\angle B_{1} \approx 31.668^{\circ}$, $\angle B_{2} \approx 148.332^{\circ}$)

$\angle C_{1} = 180^{\circ}-\angle A - \angle B_{1}$

$\angle C_{1} = 180^{\circ}-21^{\circ}-31.668^{\circ}$

$\angle C_{1} \approx 127.332^{\circ}$

$\angle C_{2} = 180^{\circ}-\angle A - \angle B_{2}$

$\angle C_{2} = 180^{\circ}-21^{\circ}-148.332^{\circ}$

$\angle C_{2} \approx 10.668^{\circ}$

Lastly, the remaining side of the triangle is found by means of the Law of Sine: ($a = 71$, $\angle A = 21^{\circ}$, $\angle C_{1} \approx 127.332^{\circ}$, $\angle C_{2} \approx 10.668^{\circ}$)

$c_{1} = a\cdot \left(\frac{\sin C_{1}}{\sin A} \right)$

$c_{1} = 71\cdot \left(\frac{\sin 127.332^{\circ}}{\sin 21^{\circ}} \right)$

$c_{1} \approx 157.532$

$c_{2} = a\cdot \left(\frac{\sin C_{2}}{\sin A} \right)$

$c_{2}= 71\cdot \left(\frac{\sin 10.668^{\circ}}{\sin 21^{\circ}} \right)$

$c_{2}\approx 36.676$

The answer are presented below:

$\angle B_{1} \approx 31.668^{\circ}$, $\angle B_{2} \approx 148.332^{\circ}$

$\angle C_{1} \approx 127.332^{\circ}$, $\angle C_{2} \approx 10.668^{\circ}$

$c_{1} \approx 157.532$, $c_{2}\approx 36.676$