Questions: Solve for the remaining angles and side of the one triangle that can be created. Round to the nearest hundredth: B=125°, c=3, b=6.5 C= A= a=

Solution Steps

To solve for the remaining angles and side of the triangle, we can use the Law of Sines and the fact that the sum of the angles in a triangle is 180 degrees. First, we can find angle \( C \) using the Law of Sines, then find angle \( A \) by subtracting the sum of angles \( B \) and \( C \) from 180 degrees. Finally, we can use the Law of Sines again to find side \( a \).

We are given the following values for the triangle:

• \( B = 125^\circ \)
• \( c = 3 \)
• \( b = 6.5 \)

Using the Law of Sines, we can find angle \( C \): \[ \frac{c}{\sin C} = \frac{b}{\sin B} \] Rearranging gives: \[ \sin C = \frac{c \cdot \sin B}{b} \] Substituting the known values: \[ \sin C = \frac{3 \cdot \sin(125^\circ)}{6.5} \] Calculating this yields: \[ C \approx 22.21^\circ \]

Using the fact that the sum of angles in a triangle is \( 180^\circ \): \[ A = 180^\circ - B - C \] Substituting the values: \[ A = 180^\circ - 125^\circ - 22.21^\circ \approx 32.79^\circ \]

Using the Law of Sines again to find side \( a \): \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Rearranging gives: \[ a = \frac{b \cdot \sin A}{\sin B} \] Substituting the known values: \[ a = \frac{6.5 \cdot \sin(32.79^\circ)}{\sin(125^\circ)} \approx 4.3 \]

Final Answer