Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Trigonometry → Applications of Trigonometry ↓


Bearings and Navigation


Directions and navigation are essential applications of trigonometry, which are particularly useful in guiding how we navigate around the world, whether on land, at sea, or in the air. Understanding directions involves understanding direction and angle measurements from a certain direction, usually north, helping in a variety of scenarios such as sailing, flying, hiking, or even finding directions in your car.

Understanding bearings

Bearings are a way of expressing the relative direction from one point to another, using a circle with 360 degrees. In navigation, bearings are usually measured in degrees clockwise from north.

For example, 0° direction points directly north, while 90° direction points directly east. 180° direction points directly south, and a 270° direction points directly west. This system allows for precise navigation, helping the navigator to determine and follow a route.

Some specific examples include:

  • 45°: North East direction
  • 135°: South-East direction
  • 225°: South-West direction
  • 315°: North-West direction

Illustrative examples of bearings

Let us look at some examples to understand this concept better:

        
            
            
            
            
            
            0° (North)
            90° (East)
            180° (South)
            270° (West)

The above image shows a compass with directions for the cardinal directions (North, East, South and West) and different lines used to indicate directions at 0°, 90°, 180° and 270°

Calculation of bearings

To calculate the direction between two points on the map, you need to know the starting and ending points. Let's consider a simple example. Let's say you are at point A and want to go to point B. This process includes:

  1. Determine the north line from your starting point.
  2. Measure the angle in clockwise direction from the north line to the line joining points A and B.

Suppose point A is at coordinates (3, 4) and point B is at (7, 8). The step-by-step calculation will involve determining the angle using trigonometric functions.

Example calculation

        1. Calculate the difference in the x-coordinates: ΔX = 7 - 3 = 4. 2. Calculate the difference in the y-coordinates: ΔY = 8 - 4 = 4. 3. Use the tangent function to find the angle θ from the north line: θ = arctan(ΔY / ΔX) = arctan(4 / 4). 4. Simplify to find θ = arctan(1) = 45°. 5. The bearing from point A to point B is: (0° + 45°) = 45°.

Thus, point B is in the direction of 45° from point A, which corresponds to the north-east direction.

Examples in real-life context

Imagine a plane is flying from a city located at (0, 0) to another city located at (5, 10). How do we calculate the direction?

        1. ΔX = 5 - 0 = 5. 2. ΔY = 10 - 0 = 10. 3. Use the tangent function: θ = arctan(10 / 5) = arctan(2). 4. Look up or use a calculator for arctan(2) ≈ 63.43°. 5. Bearing from start to destination = (0° + 63.43°) = 63.43°.

This calculation indicates the direction the pilot must take to reach the destination city from the center of the origin city, which is calculated to be a direction of approximately 63.43°.

Navigation using bearings

Navigation generally involves using a compass to figure out directions to guide yourself from one point to another. In earlier times, sailors and explorers relied heavily on directions and compasses for route-finding. Nowadays, satellites and GPS systems have simplified the process, but the basics of directions remain the same.

Following routes

When following a route, a navigator often needs to follow a series of directions to reach a destination. For example, a hiker may need to proceed through a set of directions on a trail: 45° for the first 2 kilometers, then 135°, and so on.

To understand this, consider this example:

        
            
            
            
            45°
            
            
            
            135°

The example shows movement between points using specific directions. The lines indicate the direction of travel along with the angles given at the key points, which is part of the navigation process.

Explaining bearings in everyday life

In our daily lives, we may not always realize it, but we use bearings often. Consider driving directions: "Move northeast," or "Keep driving south until you reach a landmark." These depend on bearings.

Example of vehicle navigation

Imagine you are driving from the North Pole to a city located (metaphorically) slightly to the east. Basic navigation instructions involve explaining how directions change as you progress on your journey.

Guidelines aren't just about math; they connect our understanding to the physical world, and provide guidance for navigation no matter the context.

Bearings in technology

Technology, particularly in the form of GPS systems, uses bearings in addition to other tools to accurately track and navigate positions on a global scale. Understanding bearings helps to understand part of the technology that guides planes flying across continents and ships navigating across wide oceans. They are an integral part of the algorithms that allow navigation systems to calculate the shortest paths and directions.

Role in GPS and mapping applications

Many modern mapping applications include directions as an essential part of route guidance. Algorithms calculate instantaneous directions that match the correct direction based on the current location and end point using real-time data.

For example, if you are using a GPS service and your destination is South-East of your location, it dynamically calculates the required direction to ensure you are moving accurately in that direction, and getting from point A to point B efficiently.

Mathematical model

In technology systems that require greater accuracy, models often break down complex, curved paths into smaller linear segments with defined bearings, thereby reducing navigation error. These advanced navigational algorithms consider the curvature of the Earth and integrate calculus into the process mathematics, which is further tuned by trigonometric principles, including bearings.

As is obvious, bearings are a vital cornerstone of practical trigonometry, which goes beyond theoretical diagrams to everyday navigation, whether on your smartphone or out at sea. Understanding these concepts effectively means understanding how fundamental directions also guide ventures, crafts, and adventures around the world.

Conclusion

Bearings are vital to navigation, providing a practical application of trigonometry that allows for pinpoint accuracy in understanding and directing movement. Whether it’s exploring new lands, following a GPS route, or navigating the oceans, bearings provide the directional vocabulary needed to get from one point to another. As technologies advance and our world becomes more interconnected, a fundamental understanding of bearings remains as relevant as ever, guiding us through countless journeys with accuracy and clarity.


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Bearings and Navigation

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