triangle law of sines calculation
Check the final answer first, then review the worked steps.
Problem
triangle law of sines calculation
Answer
40.1
Step-by-step solution
- Find the measure of angle C: The sum of the angles in any triangle is $180^\circ$. Given $m\angle A = 47^\circ$ and $m\angle B = 68^\circ$, we can find $m\angle C$ as follows: $m\angle C = 180^\circ - (47^\circ + 68^\circ) = 180^\circ - 115^\circ = 65^\circ$.
- Set up the Law of Sines: The Law of Sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. We are given $b = 41$, $m\angle B = 68^\circ$, and we found $m\angle C = 65^\circ$. We need to solve for $c$ using the ratio: $\frac{b}{\sin B} = \frac{c}{\sin C}$.
- Substitute the known values: Plugging in the values, we get: $\frac{41}{\sin 68^\circ} = \frac{c}{\sin 65^\circ}$.
- Solve for c: Multiply both sides by $\sin 65^\circ$ to isolate $c$: $c = \frac{41 \cdot \sin 65^\circ}{\sin 68^\circ}$.
- Calculate the final value: Using a calculator, $\sin 65^\circ \approx 0.9063$ and $\sin 68^\circ \approx 0.9272$. Therefore, $c \approx \frac{41 \cdot 0.9063}{0.9272} \approx \frac{37.1583}{0.9272} \approx 40.075$. Rounding to the nearest tenth, we get $c \approx 40.1$.