Questions: If c=181, angle A=75° and angle B=49°, b= ; Assume angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c.

Transcript text: If $c=181, \angle A=75^{\circ}$ and $\angle B=49^{\circ}$, $b=$ $\square$ ; Assume $\angle A$ is opposite side $a, \angle B$ is opposite side $b$, and $\angle C$ is opposite side $c$.

Solution

Solution Steps

To find the length of side $ b $ in a triangle where angles $ A $, $ B $, and $ C $ are given along with side $ c $, we can use the Law of Sines. The Law of Sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. First, calculate angle $ C $ using the fact that the sum of angles in a triangle is $ 180^\circ $. Then, apply the Law of Sines to solve for $ b $.

Step 1: Calculate Angle $ C $

Using the property that the sum of angles in a triangle is $ 180^\circ $, we find angle $ C $ as follows: \[ C = 180^\circ - A - B = 180^\circ - 75^\circ - 49^\circ = 56^\circ \]

Step 2: Convert Angles to Radians

To apply the Law of Sines, we convert angles $ A $, $ B $, and $ C $ from degrees to radians: \[ A_{\text{rad}} = \frac{75 \pi}{180} \approx 1.3090, \quad B_{\text{rad}} = \frac{49 \pi}{180} \approx 0.8552, \quad C_{\text{rad}} = \frac{56 \pi}{180} \approx 0.9774 \]

Step 3: Apply the Law of Sines

Using the Law of Sines: \[ \frac{b}{\sin B} = \frac{c}{\sin C} \] we can solve for $ b $: \[ b = \frac{c \cdot \sin B}{\sin C} = \frac{181 \cdot \sin(49^\circ)}{\sin(56^\circ)} \] Calculating this gives: \[ b \approx 164.7723 \]