Questions: Solve the triangle. A=39°, B=64°, a=5 C=° b ≈ (Do not round until the final answer. Then round to the nearest tenth as needed.) c ≈ (Do not round until the final answer. Then round to the nearest tenth as needed.)

Solution Steps

To solve the triangle, we need to find the missing angle $C$ and the sides $b$ and $c$.

1. Find angle $C$: Use the fact that the sum of the angles in a triangle is $180^\circ$.
2. Find side $b$: Use the Law of Sines, which states $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$.
3. Find side $c$: Again, use the Law of Sines, $\frac{a}{\sin(A)} = \frac{c}{\sin(C)}$.

Using the triangle angle sum property, we have: $$C = 180^\circ - A - B = 180^\circ - 39^\circ - 64^\circ = 77^\circ$$

Applying the Law of Sines: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$$ Substituting the known values: $$\frac{5}{\sin(39^\circ)} = \frac{b}{\sin(64^\circ)}$$ Solving for $b$: $$b = 5 \cdot \frac{\sin(64^\circ)}{\sin(39^\circ) } \approx 7.1$$

Again using the Law of Sines: $$\frac{a}{\sin(A)} = \frac{c}{\sin(C)}$$ Substituting the known values: $$\frac{5}{\sin(39^\circ)} = \frac{c}{\sin(77^\circ)}$$ Solving for $c$: $$c = 5 \cdot \frac{\sin(77^\circ)}{\sin(39^\circ)} \approx 7.7$$

Final Answer