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