Question

Solve the triangle.

B = 73°, b = 15, c = 8

Help me please

Answer 1

Answer on Question #42402 – Math - Geometry

Problem

Solve the triangle.

$$B = 73{}^\circ, \ b = 15, \ c = 8$$

Help me please

Solution.

The Law of Sines gives $\frac{\sin B}{b} = \frac{\sin C}{c} \Rightarrow \sin C = \sin B \times \frac{C}{b} = \sin 73{}^\circ \times \frac{8}{15} \approx 0.9563 \times \frac{8}{15} \approx 0.5100$.

Then we have 2 cases:

$C = \sin^{-1} 0.5100 \approx 30.6638{}^\circ$ or $C = 180{}^\circ - \sin^{-1} 0.5100 \approx 180{}^\circ - 30.6638{}^\circ = 149.3362{}^\circ$.

In the second case, $B + C = 149.3362{}^\circ + 73{}^\circ = 222.3362{}^\circ > 180{}^\circ = A + B + C$, contradiction.

Thus, $C = 30.6638{}^\circ$. Since $A + B + C = 180{}^\circ$, we have $A = 180{}^\circ - 73{}^\circ - 30.6638{}^\circ = 76.3362{}^\circ$.

Then, from the Law of Sines,

$$\frac{\sin B}{b} = \frac{\sin A}{a} \Rightarrow a = \sin A \frac{b}{\sin B} = \sin 76.3362{}^\circ \frac{15}{\sin 73{}^\circ} \approx 0.9717 \frac{15}{0.9563} \approx 15.2416.$$

So, the answer is $C = 30.6638{}^\circ$, $A = 76.3362{}^\circ$, $a = 15.2416$.

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