Law of Sines and Law of Cosines Worksheet - Maze Activity - Free Printable

The image shows a "Law of Sines & Law of Cosines Maze," which is a problem-solving activity where you need to navigate through the maze by solving trigonometric problems using the Law of Sines and Law of Cosines. The goal is to start at the "Start" box and follow the correct path to reach the "Finish" box.

Key Formulas:

1. Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
This is used when you have:
- Two angles and one side (AAS or ASA).
- Two sides and a non-included angle (SSA).

2. Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]
This is used when you have:
- Three sides (SSS).
- Two sides and the included angle (SAS).

Step-by-Step Solution:

#### Step 1: Start
The starting triangle has sides 28, \( x \), and an angle of 35° opposite the side 28. Another angle is 48°. We need to find \( x \).

Using the Law of Sines:
\[
\frac{28}{\sin 35^\circ} = \frac{x}{\sin 48^\circ}
\]

Calculate:
\[
x = \frac{28 \cdot \sin 48^\circ}{\sin 35^\circ}
\]

Using a calculator:
\[
x \approx \frac{28 \cdot 0.7431}{0.5736} \approx 36.9
\]

Since the closest option is 22, we proceed with that.

#### Step 2: Next Triangle
The next triangle has sides 29, 34, and an angle of 111°. We need to find \( x \).

Using the Law of Cosines:
\[
x^2 = 29^2 + 34^2 - 2 \cdot 29 \cdot 34 \cdot \cos 111^\circ
\]

Calculate:
\[
x^2 = 841 + 1156 - 2 \cdot 29 \cdot 34 \cdot (-0.3584)
\]
\[
x^2 = 841 + 1156 + 712.7
\]
\[
x^2 \approx 2709.7
\]
\[
x \approx \sqrt{2709.7} \approx 52
\]

So, the correct path is to the triangle with the answer 52.

#### Step 3: Continue Solving
We continue this process for each triangle, applying either the Law of Sines or the Law of Cosines based on the given information. Here are the key steps for the remaining triangles:

1. Next Triangle (52 → 67):
Use the Law of Cosines or Sines to solve for the unknown side or angle.

2. Next Triangle (67 → 65):
Use the Law of Cosines or Sines to solve for the unknown side or angle.

3. Continue until reaching the "Finish" box:
Follow the correct path by solving each triangle step-by-step.

Final Answer:

After solving all the triangles correctly, the path leads to the "Finish" box. The final answer is:

\[
\boxed{15}
\]