solve for sides of a triangle using law of sines
Check the final answer first, then review the worked steps.
Problem
solve for sides of a triangle using law of sines
Answer
\(a = 16.23, b = 17.99\)
Step-by-step solution
- Identify the given information: We are given a triangle with side $c = 9.3$, angle $A = 64^\circ$, and angle $B = 85^\circ$. We need to find the side lengths $a$ and $b$.
- Calculate the third angle C: The sum of angles in a triangle is $180^\circ$. Therefore, $C = 180^\circ - (64^\circ + 85^\circ) = 180^\circ - 149^\circ = 31^\circ$.
- Use the Law of Sines: The Law of Sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
- Calculate side a: Using $\frac{a}{\sin 64^\circ} = \frac{9.3}{\sin 31^\circ}$, we get $a = \frac{9.3 \cdot \sin 64^\circ}{\sin 31^\circ}$. Calculating this: $a \approx \frac{9.3 \cdot 0.8988}{0.5150} \approx 16.23$.
- Calculate side b: Using $\frac{b}{\sin 85^\circ} = \frac{9.3}{\sin 31^\circ}$, we get $b = \frac{9.3 \cdot \sin 85^\circ}{\sin 31^\circ}$. Calculating this: $b \approx \frac{9.3 \cdot 0.9962}{0.5150} \approx 17.99$.
- Conclusion: The calculated values are $a \approx 16.23$ and $b \approx 17.99$, which corresponds to option C.