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


Welcome to the fascinating world of the application of trigonometry known as “height and distance”. This concept is an interesting use of trigonometric ratios, which we use to measure the height of an object or the distance between two points, especially when direct measurement is not possible.

Imagine you are standing and looking at a tall building or mountain. If you want to calculate its height, can you measure it using just a ruler? No, absolutely not. This is where trigonometry helps you. By measuring the angle of elevation (or depression) and the distance from the observation point to the base of the object, it is possible to calculate unknown distances or heights.

Basic trigonometric ratios

Before we discuss heights and distances, it's important to remember some basic trigonometric concepts:

  • Sine (&sin;): Represents the ratio of the opposite side and the hypotenuse in a right-angled triangle.
  • Cosine (&cos;): Represents the ratio of the adjacent side and the hypotenuse.
  • Tangent (&tan;): Represents the ratio of the adjacent side to the opposite side.

Key terms in heights and distances

Sight

It is a straight line drawn from the eye of the observer to the object being observed.

Elevation angle

When an observer looks at an object upwards, the angle formed between the line of sight and the horizontal is called the angle of elevation.

Angle of elevation, θ = tan-1 (opposite/adjacent)
Height distance supervisor θ

Angle of depression

In contrast, the angle of depression is the angle formed when an observer looks at an object located below from a higher place.

Angle of depression, θ = tan-1 (opposite/adjacent)
Height distance supervisor θ

Heights and distances applications

Knowing how to calculate height and distance is useful in real-world scenarios, especially in fields like surveying, navigation, architecture, forestry, and even aviation. Let's take a look at some examples to strengthen our understanding.

Example 1: Calculating the height of a building

Suppose you are standing 50 m away from a building. You measure the angle of elevation from the top of the building as 30 degrees. To find the height of the building, use the tangent ratio:

tan(θ) = opposite/adjacent

For 30 degrees, tan(30) = height/50 

Height = tan(30) × 50 

Let's do the calculation:

tan(30) ≈ 0.577

Height = 0.577 × 50 = 28.85 m

Therefore, the height of the building is approximately 28.85 metres.

Example 2: Calculating distance to an object

Suppose you are standing on top of a lighthouse and looking at a boat at sea. The height of the lighthouse is 90 m, and the angle of depression to the boat is 45 degrees. Calculate how far the boat is from the base of the lighthouse.

tan(θ) = opposite/adjacent

For 45 degrees, tan(45) = 90/distance 

Distance = 90/tan(45)

Since tan(45) = 1:

Distance = 90 meters

Therefore, the boat is 90 m away from the base of the lighthouse.

Step-by-step approach

Repeating the same steps for problems involving height and distance achieves consistency and accuracy.

  1. Identify the right triangle in the problem.
  2. Determine the known sides and angles of the triangle.
  3. Select the correct trigonometric ratio to use (sine, cosine or tangent).
  4. Solve for the unknown side or angle using the trigonometric equation.
  5. Perform calculations to find the height or distance as required.

Example 3: Calculating the height of a kite

You are flying a kite and standing 100 m away from the point where the string meets the ground. The string makes an angle of 60 degrees with the ground. Calculate the height of the kite.

sin(θ) = opposite/hypotenuse

Here, hypotenuse = 100 (length of the string)
Angle θ = 60 degrees

sin(60) = height/100

Height Solution:

Height = sin(60) × 100 

Since sin(60) ≈ 0.866, 

Height = 0.866 × 100 = 86.6 m

Therefore, the height of the kite from the ground is 86.6 m.

Conclusion

The concepts of height and distance demonstrate how the world around us can be understood using the mathematical tool of trigonometry. Armed with a few angles and a few distances, you can unlock the height of mountains or the expanse across which a lighthouse shines. With practice, it becomes intuitive to interpret everyday landscapes as height and distance problems. You have armed yourself with a skill that enhances your ability to accurately measure and understand the large scale of the world.


Grade 10 → 4.4


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Trigonometric Ratios of Complementary Angles
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Angle of Elevation and Depression

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Heights and Distances

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