Question

problem #2:
solve triangle abc if a = 65°, b = 65°, and c = 6. if no triangle exists, explain why.

Explanation:

Step1: Calculate angle C

The sum of angles in a triangle is $180^\circ$.
$C = 180^\circ - A - B = 180^\circ - 65^\circ - 65^\circ = 50^\circ$

Step2: Find side a via Law of Sines

Law of Sines: $\frac{a}{\sin A} = \frac{c}{\sin C}$
$a = \frac{c \cdot \sin A}{\sin C} = \frac{6 \cdot \sin 65^\circ}{\sin 50^\circ}$
Calculate values: $\sin 65^\circ \approx 0.9063$, $\sin 50^\circ \approx 0.7660$
$a \approx \frac{6 \cdot 0.9063}{0.7660} \approx 7.09$

Step3: Find side b via Law of Sines

Since $A=B$, $a=b$ (isosceles triangle property), or use Law of Sines:
$b = \frac{c \cdot \sin B}{\sin C} = \frac{6 \cdot \sin 65^\circ}{\sin 50^\circ} \approx 7.09$

Answer:

$C = 50^\circ$, $a \approx 7.09$, $b \approx 7.09$