Solve Word Problem with Bearings | Law of Sines AAS

Summary

The script is a detailed instructional walkthrough for solving a navigation problem involving bearings and distances. It begins with a boat's departure from point A, proceeds to explain how to navigate using bearings (south 53 degrees east and south 39 degrees west) to reach points B and C, and discusses the geometric relationships and angles involved. The instructor emphasizes the importance of understanding alternate interior angles and complementary angles in determining unknown angles within a triangle. The script then transitions into using the law of sines to calculate unknown distances (d1 and d2), highlighting the need for precision and avoiding rounding errors in calculations. The goal is to find the total distance traveled by the boat.

Takeaways

• 🚤 The problem involves a boat starting at point A and moving in a specific bearing direction.
• 🧭 Bearings are used to determine the direction of travel, involving cardinal directions and angles.
• 📍 The boat first travels south 53 degrees east to point B, which is a calculated bearing from point A.
• 🛤 The second leg of the journey involves a bearing of south 39 degrees west to point C, which is 8 miles directly south of point A.
• 🔢 The task is to find the total distance traveled, which includes two unknown distances, d1 and d2.
• 📐 The problem requires understanding of geometric relationships, specifically alternate interior angles and complementary angles.
• 🔄 The script emphasizes the importance of correctly identifying and using angles in the context of parallel lines and transversals.
• 📈 The law of sines is mentioned as a method to solve for unknown distances in a non-right triangle, which is an oblique triangle in this case.
• ⚖️ The script describes setting up a ratio using the law of sines to find the unknown distances, emphasizing the importance of using exact values rather than calculated approximations.
• 🔢 The calculation involves using the sine of known angles and a known distance to find the unknown distances d1 and d2.
• 📝 The final step is to add the known distance and the two calculated distances to find the total distance traveled by the boat.

Q & A

What is the starting point of the boat's journey as described in the script?

The starting point of the boat's journey is labeled as Point A.

What is the significance of establishing cardinal directions when discussing bearings?

Establishing cardinal directions is important when discussing bearings because it helps to define the exact direction of travel relative to the cardinal points (North, East, South, West).

What does the term 'bearing' refer to in the context of the script?

In the script, 'bearing' refers to the direction in which the boat is traveling, measured in degrees from the cardinal directions.

What is the first direction the boat travels after starting from Point A?

The first direction the boat travels is South 53 degrees East.

How is Point B determined in the script?

Point B is determined by the boat traveling in the direction of South 53 degrees East for a certain distance.

What is the second direction the boat travels after reaching Point B?

The second direction the boat travels is South 39 degrees West.

Why is Point C significant in the script?

Point C is significant because it lies eight miles directly south of Point A, which is a key piece of information for calculating distances.

What geometric relationships are used to determine the angles within the triangle formed by Points A, B, and C?

The geometric relationships used include complementary angles, alternate interior angles, and the fact that the sum of angles in a triangle is 180 degrees.

Why is the Law of Sines mentioned in the script for solving the triangle?

The Law of Sines is mentioned because it is a mathematical law that can be used to find unknown sides or angles in a non-right triangle, which is the case in the script.

What is the importance of avoiding the use of calculated answers in further calculations as mentioned in the script?

Avoiding the use of calculated answers in further calculations is important to prevent the accumulation of rounding errors and to ensure the accuracy of the final results.

How does the script suggest handling the calculation of total distance traveled by the boat?

The script suggests calculating the individual distances (d1 and d2) using the Law of Sines and then summing them up along with the known distance of 8 miles to find the total distance traveled.