Maths Help: Bearing Problems - VividMath.com

VividMath
14 May 201207:04
EducationalLearning
32 Likes 10 Comments

TLDRThe video script emphasizes the importance of diagrammatic representation in solving mathematical problems, particularly in trigonometry. It illustrates this concept through a problem involving a ship's bearing and distance traveled. The speaker begins by stating that a well-drawn problem is halfway to being solved. They then guide viewers through constructing a diagram for a given scenario where a ship sails on a bearing of 200 degrees and is 18 nautical miles further west from a certain point. By identifying the right triangle formed and calculating the necessary angle, the presenter uses trigonometry to find the distance to the ship, which is the unknown variable in the problem. They demonstrate solving for the missing length using cosine, resulting in the ship being 233.9 nautical miles away. The video concludes with a call to action to visit the website for additional resources and subscribe for further assistance with math questions.

Takeaways
  • 📐 **Importance of Diagrams**: The speaker emphasizes that a well-drawn problem is halfway solved, highlighting the significance of visual representation in problem-solving.
  • 🧭 **Bearing as a Starting Point**: The script begins with a ship's bearing of 200 degrees, which sets the direction for the problem-solving process.
  • 📍 **Understanding Coordinates**: The location of point O and its relation to point P, 18 nautical miles west, is crucial for setting up the problem's spatial context.
  • 🔍 **Identifying Missing Variables**: The script introduces 'X' as the unknown distance the ship has sailed, which becomes the focus of the solution.
  • 📐 **Constructing a Triangle**: The information given allows for the creation of a right-angled triangle, which is fundamental for applying trigonometric functions.
  • 🔢 **Calculating Angles**: The angle QOP is identified as 70 degrees by subtracting the bearing (200 degrees) from the total westward angle (270 degrees).
  • 📐 **Applying Trigonometry**: The cosine function is used to relate the adjacent side (80 nautical miles) to the hypotenuse (X, the distance to be found).
  • ✍️ **Setting Up the Equation**: The script forms the equation cos(70°) = 80/X to solve for the unknown distance X.
  • 🧮 **Solving for X**: By isolating X and using a calculator, the script finds the ship's distance to be approximately 233.9 nautical miles.
  • 🌐 **Additional Resources**: The speaker encourages checking out their website for more resources, cheat sheets, transcripts, and exclusive math videos.
  • 📚 **Subscription Benefits**: Subscribing to the website offers access to ready answers for math questions and other special offers.
Q & A

  • What is the key statement shared in the transcript regarding problem-solving in trigonometry and math?

    -The key statement is 'A problem well drawn is a problem half solved.' This emphasizes the importance of visual representation in understanding and solving mathematical problems.

  • What is the bearing direction mentioned for the ship in the example problem?

    -The bearing direction mentioned for the ship is 200 degrees.

  • How far west from point O is point P in the given problem?

    -Point P is 18 nautical miles further west from point O.

  • What is the missing variable in the problem that represents the distance the ship has sailed?

    -The missing variable in the problem is represented by 'X', which is the distance from the starting point O to the ship's location.

  • What type of triangle is formed in the diagram?

    -A right-angled triangle is formed in the diagram, which is useful for applying trigonometric functions to find the missing distance 'X'.

  • What trigonometric function is used to find the missing distance 'X' in the triangle?

    -The cosine function is used to find the missing distance 'X' because it relates the adjacent side (80 nautical miles) to the hypotenuse (the distance 'X').

  • What is the angle calculated to be in the right-angled triangle?

    -The angle calculated in the right-angled triangle is 70 degrees, which is found by subtracting the bearing (200 degrees) from the total westward angle (270 degrees).

  • How is the equation set up to solve for the missing distance 'X'?

    -The equation is set up as Cos(70 degrees) = 80 / X, where 80 is the length of the adjacent side, and 'X' is the hypotenuse representing the distance the ship has sailed.

  • What is the process to isolate 'X' and find its value?

    -To isolate 'X', both sides of the equation are multiplied by 'X' and then divided by cos(70 degrees), which gives X = 80 / cos(70 degrees).

  • What is the calculated value of 'X' using a calculator?

    -The calculated value of 'X' is approximately 233.9 nautical miles.

  • What additional resources are available for those interested in further math help?

    -Additional resources, including cheat sheets, transcripts, and other math videos, can be found on the website http://www.vividmaths.com. Subscribers also get access to ready answers for math questions and special offers.

  • What is the final step in the process of solving the problem as described in the transcript?

    -The final step is to use a calculator to divide 80 by the cosine of 70 degrees to find the value of 'X', which represents the distance the ship has sailed.

Outlines
00:00
📐 Introduction to Problem-Solving with Diagrams in Trigonometry

The speaker begins by emphasizing the importance of diagrammatic representation in solving mathematical problems, particularly trigonometry. They introduce a specific problem involving a ship's bearing and distances. The process starts with understanding the bearing, which in this case is 200 degrees from a starting point 'O'. The speaker then guides the audience through visualizing the problem by drawing a line representing the bearing and marking an additional 18 nautical miles to the west, indicating the position 'P'. The objective is to find the distance 'X' from the starting point 'O' to the ship's location. The key takeaway is that a well-drawn diagram is crucial for solving the problem, and the speaker proceeds to the next step, which involves calculating the necessary angles to find the missing distance 'X'.

05:02
🧮 Applying Trigonometry to Find the Missing Distance

The second paragraph delves into the trigonometric calculations needed to find the missing distance 'X'. The speaker identifies a right-angled triangle formed by the bearing line, the additional westward distance, and the hypotenuse representing the ship's location. They calculate the angle QOP by subtracting the bearing (200 degrees) from the total westward angle (270 degrees), which results in 70 degrees. With the angle and the adjacent side (80 nautical miles) known, the speaker uses the cosine function to establish the relationship between the angle, the adjacent side, and the hypotenuse (missing distance 'X'). The equation cos(70 degrees) = 80/X is formed, and by solving for X, the speaker finds the missing distance to be approximately 233.9 nautical miles. The speaker concludes with a call to action to visit their website for additional resources and to subscribe for further assistance with math problems.

Mindmap
Keywords
💡Bearing
Bearing refers to the direction or orientation of a ship or aircraft relative to a reference point. In the video, it is used to establish the initial direction of the ship's journey, which is 200 degrees from a starting point, or 'O'. This is a fundamental concept in the problem-solving process as it sets the stage for the subsequent geometric and trigonometric calculations.
💡Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. The video emphasizes the importance of trigonometry in solving the problem by using it to find the unknown distance (X) that the ship has sailed, by applying trigonometric ratios to a right-angled triangle formed by the ship's path.
💡Diagram
A diagram is a visual representation used to illustrate and solve problems. The video script stresses the importance of diagramming as it helps to visualize the problem and find a solution. In this context, the diagram is essential for translating the textual information about the ship's bearing and distance into a geometric form that can be analyzed mathematically.
💡Nautical Miles
Nautical miles are a unit of distance used in navigation and aviation. One nautical mile is equivalent to exactly 1.852 kilometers. In the video, the script uses nautical miles as the unit of measurement for the distance the ship travels, which is stated as 18 nautical miles further west from the starting point.
💡Right-Angle Triangle
A right-angle triangle is a triangle in which one of the angles is 90 degrees. The video script identifies the geometric shape formed by the ship's path and uses it to apply trigonometric functions. The right-angle triangle is crucial in this context as it allows for the application of trigonometric ratios to find the unknown side lengths.
💡Trigonometric Ratios
Trigonometric ratios, such as sine, cosine, and tangent, are used to relate the angles of a right-angled triangle to the ratios of its sides. In the video, the cosine ratio is used to find the missing distance (X) that the ship has sailed, by setting up an equation involving the known adjacent side (80 nautical miles) and the angle (70 degrees).
💡Problem-Solving
Problem-solving is the process of finding solutions to problems. The video script outlines a method for solving mathematical problems, particularly those involving geometry and trigonometry. It emphasizes the importance of visualizing the problem through a diagram and then applying mathematical concepts to find the solution.
💡Port
A port is a location on a coast or shore containing one or more harbors where ships can dock and transfer people or cargo. In the video, the port is mentioned as the starting point ('O') from which the ship sails on a bearing of 200 degrees, setting the context for the direction and distance calculations that follow.
💡Missing Pronumeral
A pronumeral, or variable, is a symbol that stands for an unknown quantity in a mathematical problem. In the video, 'X' is used as the missing pronumeral to represent the unknown distance the ship has sailed. The process of finding the value of 'X' is central to solving the problem presented in the video.
💡Adjacent
Adjacent in geometry refers to a side that is next to another side, particularly in the context of angles. In the video, the term is used to describe the relationship between the known side of 80 nautical miles and the angle of 70 degrees, which helps in determining the correct trigonometric ratio (cosine) to use for solving the problem.
💡Hypotenuse
The hypotenuse is the longest side of a right-angled triangle, which is opposite the right angle. In the video, the hypotenuse is the unknown distance (X) that the ship has sailed. The hypotenuse is a key component in the application of trigonometric ratios, as it is the side that is being solved for in this problem.
💡Calculator
A calculator is an electronic device used to perform arithmetic operations and mathematical calculations. In the video, a calculator is mentioned as the tool used to divide 80 by the cosine of 70 degrees to find the value of the missing pronumeral (X), which is the distance the ship has sailed.
Highlights

The importance of diagrammatic representation in solving math problems, particularly in trigonometry.

The concept that a well-drawn problem is halfway to being solved.

Starting with a bearing of 200 degrees to establish the initial direction in a problem.

Identifying the position of point P, which is 18 nautical miles further west from point O.

Using the information given to set up a right-angle triangle for trigonometric calculations.

Calculating the angle QOP to be 70 degrees by subtracting the bearing from the total westward angle.

Determining the appropriate trigonometric ratio (cosine) to use for solving the triangle.

Setting up the equation cos(70 degrees) = adjacent/hypotenuse to find the missing distance X.

Solving for X by isolating it on one side of the equation and using the cosine value.

Using a calculator to find the value of X, which is approximately 233.9 nautical miles.

The practical application of trigonometry in navigation and the importance of accurate calculations.

The availability of additional resources, cheat sheets, and transcripts on the website vividmaths.com.

The offer of subscription for access to ready answers and special offers for math questions.

The emphasis on the step-by-step approach to solving trigonometric problems.

The use of the cosine function to find the length of the hypotenuse in a right-angle triangle.

The process of turning verbal information into a visual diagram to facilitate problem-solving.

The mathematical transformation from a geometric diagram to an algebraic equation.

The conversion of the trigonometric equation to find the exact distance the ship has sailed.

The integration of mathematical concepts with real-world scenarios, such as nautical navigation.

Transcripts

Browse More Related Video

Sine & Cosine Rules applied to Bearings : ExamSolutions Maths Revision

How To: Bearings Problem - VividMath.com

Bearing and Distance | Lesson 1

Bearings | Distance Bearing Problems | Likely Examination Questions | Maths Center

Related Rate Problem # 2 - Aircraft Converging

Learn to find the missing angles for a triangle using inverse trig functions

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
TrigonometryVisual Problem-SolvingMathematicsBearing CalculationNautical MilesRight TriangleCosine RuleDistance FindingEducational VideoVivid MathsMath Tutorial