Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Trigonometry → Heights and Distances ↓


Applications in Real-Life Problems


Trigonometry is a fundamental aspect of mathematics that combines the study of triangles and the calculations of angles and lengths. In Class 10, students dive into the practical applications of trigonometry in solving real-life problems, focusing on heights and distances. This subject not only enhances mathematical skills but also demonstrates the usefulness of trigonometric principles in everyday life.

Understanding the basics

Before we dive deeper into the applications, it is important to understand the basic trigonometric ratios: sine, cosine, and tangent. These ratios use the sides of a right triangle. Consider a right triangle with angle θ :

sine(θ) = opposite/hypotenuse
cosine(θ) = adjacent / hypotenuse
Tangent(θ) = opposite/adjacent

These ratios allow us to find unknown heights and distances based on some initial measurements or observations.

Heights and distances

Heights and distances form the basis of many practical problem-solving scenarios. This includes calculating the height of trees or buildings, the distance of a boat from the shore, or the altitude in an airplane's flight path.

Basic example - Finding the height of a tree

D H

Suppose you want to find the height of a tree. Stand at a certain distance, d, from the base of the tree and measure the angle of elevation, which is the angle made by your line of sight with the horizontal ground as you look toward the top of the tree (angle θ ).

h = d * tan(θ)

Using this formula and your measurements, you can calculate the height h of the tree.

Practical considerations

In practice, you may not have perfect conditions. There may be inaccuracies in measuring distances or angles. Always consider these factors:

  • Make sure your distance measurement is level with the base of the object.
  • Use an accurate protractor or clinometer to measure the angle.
  • The measurement must be free from visual obstructions.

Advanced applications - Navigation

Another real-life application of heights and distances is in navigation. For example, using a lighthouse to find the distance of a ship from the shore can be illustrated using trigonometric principles.

D H θ

Suppose the ship sees the top of the lighthouse at an angle θ to the horizontal. Given the actual height H of the lighthouse, the distance D from the ship to the base of the lighthouse is found using:

d = h / tan(θ)

Shipping authorities use this principle to establish safe travel patterns for ships, helping to prevent accidents and collisions.

Real life problems in aviation

In aviation, trigonometry is used extensively to calculate the trajectory of an aircraft, determine altitude, and locate points of climb and descent. For example, the angle of elevation helps pilots determine if they have exceeded the optimal angle to save fuel.

Example - Finding the altitude of an airplane

D H θ

Consider an airplane that takes off and travels straight upward until it is visible at an angle of elevation θ from an observation point located at a distance d horizontally from the lift-off point. Here, the airplane's altitude h is determined by:

h = d * tan(θ)

This calculation helps in maintaining the desired altitude and ensuring safe travel for the passengers.

Hindering factors

However, every calculation assumes certain ideal conditions. Measurements during flight can be affected by a variety of factors such as wind flow, the curvature of the Earth, and changes in atmospheric pressure, all of which can slightly distort trigonometric calculations.

Conclusion

Understanding and applying trigonometry in real-life situations such as finding heights and distances is a vital skill. Whether it is through simple activities such as measuring the height of a tree or complex tasks such as airplane navigation, the principles of trigonometry provide the tools necessary for problem-solving. Encouragement and focus on such applications in mathematics education produces not only competent mathematicians but also practical problem-solvers who are prepared to tackle a variety of real-world challenges.


Grade 10 → 4.4.2


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Applications in Real-Life Problems

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