Questions: Solve each triangle ABC that exists. A=37.9° a=3.3 c=12

Solution Steps

To solve the triangle \(ABC\) given \(A=37.9^\circ\), \(a=3.3\), and \(c=12\), we can use the Law of Sines and the Law of Cosines. First, we can use the Law of Sines to find angle \(C\). Then, we can use the Law of Cosines to find side \(b\). Finally, we can find angle \(B\) by subtracting the sum of angles \(A\) and \(C\) from \(180^\circ\).

We are given the following information about triangle \(ABC\):

• Angle \(A = 37.9^\circ\)
• Side \(a = 3.3\)
• Side \(c = 12\)

To solve for the unknowns in the triangle, we can use the Law of Sines: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

First, we solve for \(\sin C\): \[ \sin C = \frac{c \cdot \sin A}{a} \] Substitute the given values: \[ \sin C = \frac{12 \cdot \sin 37.9^\circ}{3.3} \]

Using a calculator: \[ \sin 37.9^\circ \approx 0.6142 \]

\[ \sin C = \frac{12 \cdot 0.6142}{3.3} \approx 2.2327 \]

Since \(\sin C\) must be between -1 and 1 for a valid triangle, \(\sin C \approx 2.2327\) is not possible. This means that no such triangle exists with the given parameters.

Final Answer