Bearing and Distance | Lesson 1

Tambuwal Maths Class
12 Apr 202112:27
EducationalLearning
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TLDRThe video script presents a step-by-step guide to solving a navigation problem involving bearings and distances. A boy starts at point P and walks 6 kilometers on a bearing of 65 degrees to reach point Q. Then, he walks a further 13 kilometers on a bearing of 146 degrees to reach point R. The task is to calculate the direct distance from point P to point R. The solution involves constructing angles and using trigonometric principles, specifically the cosine rule, to find the missing side. After sketching the problem and applying the cosine rule, the calculated distance from P to R is approximately 15 kilometers. The video emphasizes the importance of geometry and trigonometry in solving bearing problems and provides a clear method for finding the direct distance between two points given the lengths of two sides and the included angle.

Takeaways
  • ๐Ÿ“ **Understanding Bearings**: The problem involves calculating distances using bearings, which are specified directions relative to true north.
  • ๐Ÿšถ **First Leg of the Journey**: The boy initially walks 6 kilometers from point P to point Q on a bearing of 65 degrees.
  • ๐Ÿ“ **Second Leg of the Journey**: From point Q, he walks 13 kilometers to point R on a bearing of 146 degrees.
  • ๐Ÿ“ **Constructing Angles**: To solve the problem, angles are constructed using the bearings provided, with the first angle being 65 degrees from P.
  • ๐Ÿ“ **Distance Calculation**: The distance from Q to R is longer, and the bearing is used to construct an angle of 146 degrees from the east.
  • ๐Ÿ“ **Sketching the Problem**: A rough sketch is made to visualize the paths and bearings, which helps in understanding the problem and applying the cosine rule.
  • ๐Ÿ”ข **Applying Cosine Rule**: The cosine rule is used to find the missing side (distance from P to R) when two sides and the included angle are known.
  • โœ… **Calculating x (Distance PR)**: Using the cosine rule, the distance from P to R is calculated by substituting the known distances and angles into the formula.
  • ๐Ÿ”ข **Final Calculation**: After applying the cosine rule and performing the calculations, the distance PR is found to be approximately 15 kilometers.
  • ๐Ÿ“ **Geometry and Trigonometry**: Knowledge of geometry and trigonometry is emphasized as crucial for solving bearing problems.
  • ๐Ÿ“ **Accuracy in Sketching**: While sketching, care is taken to ensure that the angles and distances are roughly accurate to perform the correct calculations.
  • ๐Ÿ“ˆ **Rounding the Answer**: The final answer for the distance from P to R is rounded to the nearest kilometer as per the problem's instructions.
Q & A

  • What is the first bearing the boy uses to travel from point P to point Q?

    -The first bearing is 65 degrees.

  • How far does the boy walk from point P to point Q?

    -The boy walks 6 kilometers from point P to point Q.

  • What is the bearing from point Q to point R?

    -The bearing from point Q to point R is 146 degrees.

  • How far does the boy walk from point Q to point R?

    -The boy walks 13 kilometers from point Q to point R.

  • Why is it important to construct the angles accurately when solving bearing problems?

    -Constructing the angles accurately is important because it ensures the correct direction and distance between points, which is essential for calculating the distances and positions in the problem.

  • What is the total angle between the lines from P to Q and from Q to R?

    -The total angle between the lines from P to Q and from Q to R is 99 degrees.

  • What trigonometric rule is used to find the missing side in a triangle when two sides and the included angle are known?

    -The cosine rule is used to find the missing side in such a triangle.

  • What is the formula for the cosine rule?

    -The cosine rule formula is: aยฒ = bยฒ + cยฒ - 2bc * cos(ฮธ), where a, b, and c are the sides of the triangle and ฮธ is the angle between sides b and c.

  • What is the cosine of 99 degrees?

    -The cosine of 99 degrees is approximately -0.156.

  • How do you find the distance from point P to point R (PR) after calculating x squared?

    -You find the distance PR by taking the square root of x squared.

  • What is the approximate distance from point P to point R?

    -The approximate distance from point P to point R is 15 kilometers.

  • Why is it necessary to round the final answer to the nearest kilometer?

    -Rounding to the nearest kilometer is necessary because the problem asks for the distance to be given in kilometers, and rounding gives a more practical and understandable value.

Outlines
00:00
๐Ÿ“ Introduction to the Bearing Problem

The first paragraph introduces the problem of calculating the distance from point P to point R. The problem involves a boy walking from point P to Q on a bearing of 65 degrees for 6 kilometers, and then from Q to R on a bearing of 146 degrees for 13 kilometers. The speaker emphasizes the importance of understanding bearings and constructing angles accurately. The process begins by sketching the initial bearing from P to Q, noting that all angles are measured clockwise from the north.

05:01
๐Ÿ“ Constructing the Angles and Sketching the Path

In the second paragraph, the speaker continues with the problem-solving process by sketching the path from Q to R. The sketch involves constructing an angle of 146 degrees, which is achieved by adding 56 degrees to the existing 90 degrees from the east. The sketch is described as rough, with the lengths of the paths from P to Q and Q to R being 6 and 13 kilometers, respectively. The speaker also explains the importance of including the north pole (knot) in the sketch and provides a visual description of how the angles and lines should be drawn.

10:01
๐Ÿงฎ Applying Trigonometry to Find the Distance from P to R

The third paragraph concludes the problem by calculating the distance from P to R. The speaker uses trigonometry, specifically the cosine rule, to find the missing side (distance PR). The cosine rule is applied with the known sides (6 km and 13 km) and the included angle (99 degrees). The calculation involves squaring the lengths, applying the cosine of the angle, and solving for x (the distance PR). After performing the calculations, the speaker finds that x squared equals 229.336, and by taking the square root, the distance PR is approximately 15 kilometers, rounded to the nearest kilometer.

Mindmap
Keywords
๐Ÿ’กBearing
Bearing refers to the direction or orientation of a line, often measured in degrees relative to true north. In the context of the video, bearing is crucial for understanding the path the boy takes from point P to Q and then to R. It is used to construct angles and determine the direction of travel, which is fundamental to solving the problem of finding the distance from P to R.
๐Ÿ’กSketch
A sketch in this context is a rough diagram used to visualize the problem and plan the solution. The video emphasizes the importance of sketching to understand the relative positions of points P, Q, and R, and the angles between them. The sketch helps to apply geometric principles and trigonometric calculations to find the distance from P to R.
๐Ÿ’กCosine Rule
The Cosine Rule is a fundamental principle in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. In the video, the Cosine Rule is applied to find the missing side of a triangle formed by the paths from P to Q, Q to R, and the direct line from P to R. It is essential for solving the problem as it allows calculating the distance from P to R when two sides and the included angle are known.
๐Ÿ’กTrigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. The video highlights the importance of trigonometric knowledge, particularly the use of the Cosine Rule, in solving problems involving bearings and distances. Trigonometry provides the mathematical tools necessary to calculate the unknown distance from P to R based on the given information.
๐Ÿ’กDistance Calculation
Distance calculation is the process of determining the length of a route or the space between two points. In the video, the main problem is to calculate the distance from point P to point R. This involves understanding bearings, constructing a sketch, and applying the Cosine Rule to find the direct distance between the two points, which is a central theme of the video.
๐Ÿ’กAngle Measurement
Angle measurement is the process of determining the magnitude of an angle in a geometric figure. The video script describes the need to measure angles in degrees, specifically 65 degrees and 146 degrees, to establish the bearings from points P to Q and Q to R, respectively. Accurate angle measurement is vital for constructing the correct sketch and applying trigonometric principles to find the distance from P to R.
๐Ÿ’กNorth-South-East-West (NSEW)
NSEW refers to the cardinal directions: North, South, East, and West. These are used in the video to establish a frame of reference for constructing the bearing angles and sketching the path. The cardinal points help orient the viewer and provide a common understanding of the directions in which the boy travels from P to Q and then to R.
๐Ÿ’กRough Sketch
A rough sketch is a simple, not-to-scale drawing used to illustrate a problem or concept. In the video, a rough sketch is used to visualize the path the boy takes and the angles formed between the points P, Q, and R. This sketch is essential for understanding the problem and applying the Cosine Rule to find the distance from P to R.
๐Ÿ’กDegrees
Degrees are a unit of measurement for angles, with a full circle being 360 degrees. The video script mentions specific degrees (65 degrees and 146 degrees) to describe the bearings from points P to Q and Q to R. These degree measurements are critical for constructing the angles in the sketch and for the subsequent trigonometric calculations.
๐Ÿ’กKilometers
Kilometers are a unit of length in the metric system, equivalent to 1,000 meters. The video discusses distances in kilometers, specifically the 6-kilometer walk from P to Q and the 13-kilometer walk from Q to R. These distances are key pieces of information used in the sketch and the application of the Cosine Rule to find the distance from P to R.
๐Ÿ’กNearest Kilometer
The term 'nearest kilometer' refers to rounding a numerical value to the closest whole number in kilometers. In the video, the final distance from P to R is rounded to the nearest kilometer to provide a practical and approximate measure of the distance. This rounding is a common practice when dealing with distances to make them easier to understand and communicate.
Highlights

The problem involves calculating the distance from point P to point R, given the distances and bearings between points P, Q and R.

The boy walks 6 km from P to Q on a bearing of 65 degrees.

Then he walks 13 km from Q to R on a bearing of 146 degrees.

A rough sketch is made to visualize the problem and the angles involved.

Angles are measured clockwise from north, with the north being the reference point.

The angle between the two legs PQ and QR is calculated to be 99 degrees.

The cosine rule is used to find the missing side PR, given the lengths of the other two sides and the included angle.

The formula for the cosine rule is aยฒ = bยฒ + cยฒ - 2bc cos(ฮธ).

The values are plugged into the formula to calculate PRยฒ.

The cosine of 99 degrees is approximately -0.156.

PRยฒ is calculated to be 229.336.

The square root of 229.336 is taken to find the distance PR.

The distance from P to R, PR, is approximately 15 kilometers.

The solution involves geometry and trigonometry concepts.

Constructing angles and sketching the problem is crucial for solving bearing problems.

The knowledge of geometry and trigonometry is essential for solving bearing problems.

The cosine rule is a powerful tool for finding missing sides in triangles when two sides and the included angle are known.

The solution is rounded to the nearest kilometer as requested in the problem.

Transcripts

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TrigonometryBearing CalculationGeometryDistance CalculationAnglesSketchingCosine RuleNavigationMathematicsEducational ContentProblem Solving