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Bearing Word Problem Find Distance and Angle Using Law of Sines and Law of Cosines
Mario's Math Tutoring
Overview
This video explains how to solve bearing word problems, which involve navigation and directional angles. It demonstrates a step-by-step process for finding distances and bearings between locations using the Law of Sines and the Law of Cosines. The explanation emphasizes the importance of drawing accurate diagrams, understanding how bearings are measured (clockwise from North), and utilizing geometric principles like alternate interior angles to find unknown angles within the triangle formed by the travel paths. The video walks through a specific example of a ship traveling between three cities, calculating the direct distance and bearing from the final city back to the starting point.
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Chapters
- Bearing problems involve calculating distances and directions, often using the Law of Sines and Cosines.
- Bearings are measured clockwise from North, unlike standard trigonometric angles measured counterclockwise from the positive x-axis.
- The first step is to draw a directional system (North, South, East, West) and plot the initial movement based on the given bearing and distance.
Correctly interpreting and visualizing bearings is crucial because they differ from standard angle measurements, and an accurate diagram is the foundation for solving the problem.
A ship travels 200 miles at a bearing of 50° from City A to City B. This is visualized by drawing a North line from City A and measuring 50° clockwise, then drawing a line segment 200 miles long.
- When a bearing is given as 'North 60° West', it means starting from North and rotating 60° towards the West.
- Draw a new North-South line at the destination point (City B) to plot the next leg of the journey.
- Use geometric properties, such as alternate interior angles formed by parallel North-South lines and transversals (travel paths), to find angles within the triangle.
Accurately plotting each leg of the journey and finding the internal angles of the triangle formed by the cities is essential for applying the Law of Sines and Cosines correctly.
From City B, the ship travels 150 miles at a bearing of North 60° West to City C. This involves drawing a North line at B, measuring 60° west from it, and drawing a 150-mile segment. Alternate interior angles are used to find that an angle within the triangle ABC is 40° (from the 90° angle minus the 50° bearing) and another is 30° (from the 90° angle minus the 60° bearing).
- The Law of Cosines is used when you have a Side-Angle-Side (SAS) configuration in the triangle.
- The formula is c² = a² + b² - 2ab cos(C), where C is the angle between sides a and b.
- In this problem, the sides are the distances traveled (200 miles and 150 miles), and the angle between them is the sum of the calculated internal angles (40° + 30° = 70°).
The Law of Cosines allows us to find the length of the third side of a triangle when two sides and the included angle are known, which is necessary to find the direct distance between the start and end points.
To find the distance from City C to City A (side 'x'), the Law of Cosines is applied: x² = 150² + 200² - 2(150)(200)cos(70°). Solving this gives a distance of approximately 204.9 miles.
- The Law of Sines (sin(A)/a = sin(B)/b = sin(C)/c) is used to find unknown angles when you know at least one angle and its opposite side, along with another side.
- It's used here to find an angle within the triangle that helps determine the final bearing.
- The bearing from City C to City A is calculated by finding an angle relative to North at City C and then expressing it in the standard bearing format (clockwise from North or North/South-East/West).
The Law of Sines helps find the specific angles needed to construct the final bearing, which describes the direction from the last point back to the origin.
To find the angle (let's call it 'y') opposite the 200-mile side, the Law of Sines is used: sin(y)/200 = sin(70°)/204.9. Solving for 'y' gives approximately 66.5°. This angle, combined with the previously identified 30° angle, helps determine the bearing from City C to City A.
- The calculated angle (66.5°) is part of the larger angle measured from City C.
- To find the bearing from C to A, consider the North line at C. The angle calculated using the Law of Sines (66.5°) is relative to the line segment CA.
- The final bearing can be expressed in two ways: clockwise from North, or as a direction (South) plus an angle towards West.
The final bearing provides the precise directional information needed to travel from the final destination back to the starting point, completing the problem.
The angle calculated (66.5°) is used with the 30° angle (formed by the North line at C and the line segment CB) to find the bearing from C to A. The bearing is found to be South 6.5° West, meaning from South at City C, turn 6.5° towards the West.
Key takeaways
- Bearing problems require careful visualization and drawing of directional systems.
- Bearings are measured clockwise from North, a key distinction from standard angle measurements.
- Geometric principles like alternate interior angles are vital for finding unknown angles within the travel triangle.
- The Law of Cosines is the tool for finding a side length when given two sides and the included angle (SAS).
- The Law of Sines is used to find unknown angles when given an angle and its opposite side, plus another side.
- The final bearing is determined by calculating angles relative to the North-South line at the final location.
- Problems often involve combining geometric angle properties with trigonometric laws (Sines and Cosines).
Key terms
BearingLaw of SinesLaw of CosinesAlternate Interior AnglesClockwiseCounterclockwiseSide-Angle-Side (SAS)TrigonometryNavigation
Test your understanding
- How does the measurement of a bearing differ from the measurement of a standard angle in trigonometry?
- Why is drawing an accurate diagram essential for solving bearing word problems?
- Under what triangle configuration (e.g., SAS, ASA) would you choose to use the Law of Cosines versus the Law of Sines?
- Explain how alternate interior angles can be used to find unknown angles in a bearing problem diagram.
- What information is needed to calculate the final bearing from City C back to City A?