Question

consider a triangle abc like the one below. suppose that a = 22, b = 64, and c = 44. (the figure is not drawn to scale.) solve the triangle. carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth. if there is more than one solution, use the button labeled \or\. a = □°, b = □°, c = □°

Explanation:

Step1: Use the Law of Cosines to find angle A

The Law of Cosines formula for angle \( A \) is \( \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc} \).
Substitute \( a = 22 \), \( b = 64 \), \( c = 44 \) into the formula:
\( \cos A=\frac{64^{2}+44^{2}-22^{2}}{2\times64\times44}=\frac{4096 + 1936- 484}{5632}=\frac{5548}{5632}\approx0.9851 \)
Then \( A=\arccos(0.9851)\approx9.9^{\circ} \) (calculated using a calculator, keeping intermediate steps to four decimal places)

Step2: Use the Law of Cosines to find angle C

The Law of Cosines formula for angle \( C \) is \( \cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab} \)
Substitute \( a = 22 \), \( b = 64 \), \( c = 44 \) into the formula:
\( \cos C=\frac{22^{2}+64^{2}-44^{2}}{2\times22\times64}=\frac{484 + 4096- 1936}{2816}=\frac{2644}{2816}\approx0.9389 \)
Then \( C=\arccos(0.9389)\approx20.1^{\circ} \) (calculated using a calculator, keeping intermediate steps to four decimal places)

Step3: Find angle B using the triangle angle sum

Since the sum of angles in a triangle is \( 180^{\circ} \), we have \( B = 180^{\circ}-A - C \)
Substitute \( A\approx9.9^{\circ} \) and \( C\approx20.1^{\circ} \) into the formula:
\( B=180 - 9.9-20.1 = 150.0^{\circ} \)

Answer:

\( A\approx9.9^{\circ} \), \( B = 150.0^{\circ} \), \( C\approx20.1^{\circ} \)