Question

current attempt in progress
please view the following video before answering this question. lecture video, section 4.4, example 02

note: the figure is not drawn to scale.
solve the triangle.
round the answers to the nearest integer.

Explanation:

Step1: Find the angle \( \gamma \)

The sum of angles in a triangle is \( 180^{\circ} \). So, \( \gamma=180^{\circ}-(65^{\circ} + 40^{\circ})=75^{\circ} \)

Step2: Use the Law of Sines to find \( b \)

The Law of Sines states \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). Here, \( c = 21\) cm, \( B = 40^{\circ}\), \( C=75^{\circ}\). So, \( \frac{b}{\sin40^{\circ}}=\frac{21}{\sin75^{\circ}} \), then \( b=\frac{21\times\sin40^{\circ}}{\sin75^{\circ}} \).
Using \( \sin40^{\circ}\approx0.6428\), \( \sin75^{\circ}\approx0.9659 \), \( b=\frac{21\times0.6428}{0.9659}\approx14\) cm.

Step3: Use the Law of Sines to find \( a \)

Since \( \frac{a}{\sin65^{\circ}}=\frac{21}{\sin75^{\circ}} \), then \( a=\frac{21\times\sin65^{\circ}}{\sin75^{\circ}} \).
Using \( \sin65^{\circ}\approx0.9063\), \( \sin75^{\circ}\approx0.9659 \), \( a=\frac{21\times0.9063}{0.9659}\approx20\) cm.

Answer:

\( \gamma = 75^{\circ}\), \( b = 14\) cm, \( a = 20\) cm