Some Applications of Trigonometry Class 10 Notes in Hindi (NCERT) | Chapter 9 Maths Notes PDF, Formula, Important Questions & Board Exam

यदि आप Class 10 Maths Chapter 9 – Some Applications of Trigonometry Class 10 Notes in Hindi की आसान, विस्तृत और NCERT आधारित स्टडी मटेरियल की तलाश कर रहे हैं, तो यह लेख आपके लिए है। इस अध्याय में ऊँचाई (Height), दूरी (Distance), Angle of Elevation, Angle of Depression, Line of Sight तथा उनके वास्तविक जीवन (Real-Life Applications) में उपयोग को सरल भाषा में समझाया गया है।

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Some Applications of Trigonometry Class 10 Notes: NCERT Chapter 9 Maths

Some Applications of Trigonometry is one of the most important chapters of Class 10 Mathematics. In this chapter, we use trigonometric ratios to find the height and distance of objects that cannot be measured directly. Examples include the height of a building, a tree, a tower, or the distance of a ship from a lighthouse.

इस अध्याय में हम त्रिकोणमितीय अनुपातों (Trigonometric Ratios) की सहायता से उन वस्तुओं की ऊँचाई (Height) और दूरी (Distance) ज्ञात करते हैं जिन्हें सीधे मापना कठिन होता है। यह अध्याय वास्तविक जीवन (Real Life Applications) पर आधारित है।

What is Application of Trigonometry? (त्रिकोणमिति का अनुप्रयोग क्या है?)

• Trigonometry helps us calculate unknown heights and distances.
• It is widely used in Engineering, Architecture, Surveying, Navigation and Astronomy.
• We use right-angled triangles for solving problems.
• Height and distance are calculated using trigonometric ratios like sin, cos and tan.
• Real-life examples include measuring mountains, electric poles, towers and bridges.
• इस अध्याय में मुख्य रूप से tan θ का सबसे अधिक उपयोग होता है।
• सभी प्रश्न चित्र (Diagram) बनाकर हल किए जाते हैं।

Line of Sight (दृष्टि रेखा)

Definition

The imaginary straight line joining the observer’s eye to the object is called the Line of Sight.

दर्शक (Observer) की आँख से वस्तु तक जाने वाली काल्पनिक सीधी रेखा को दृष्टि रेखा (Line of Sight) कहते हैं।

Important Points

• It is always a straight line.
• It joins the observer and the object.
• It helps in measuring angle of elevation or depression.
• बिना Line of Sight के कोण नहीं बनाया जा सकता।
• सभी प्रश्नों में यह सबसे पहले बनायी जाती है।

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Horizontal Line (क्षैतिज रेखा)

Definition

A line parallel to the ground passing through the observer’s eye is called the Horizontal Line.

दर्शक की आँख से होकर भूमि के समानांतर जाने वाली रेखा को क्षैतिज रेखा कहते हैं।

Important Points

• It is always parallel to the ground.
• It is used as the reference line.
• Angles of elevation and depression are measured from this line.
• Horizontal line हमेशा 0° दिशा को दर्शाती है।
• Diagram में इसे सीधी रेखा के रूप में बनाया जाता है।

Angle of Elevation (उन्नयन कोण)

Definition

When the observer looks upward at an object, the angle formed between the horizontal line and the line of sight is called the Angle of Elevation.

जब कोई व्यक्ति ऊपर स्थित वस्तु को देखता है, तब क्षैतिज रेखा और दृष्टि रेखा के बीच बनने वाला कोण उन्नयन कोण कहलाता है।

Important Points

• Observer looks upward.
• Angle is always above the horizontal line.
• Mostly used to calculate height.
• Angle of Elevation is always positive.
• Tower, tree and building questions generally use this angle.

Angle of Depression (अवनमन कोण)

Definition

When the observer looks downward at an object, the angle formed between the horizontal line and the line of sight is called the Angle of Depression.

जब कोई व्यक्ति नीचे स्थित वस्तु को देखता है, तब क्षैतिज रेखा और दृष्टि रेखा के बीच बनने वाला कोण अवनमन कोण कहलाता है।

Important Points

• Observer looks downward.
• Angle is measured below the horizontal line.
• Mostly used in lighthouse and aircraft problems.
• It is equal to the corresponding angle of elevation.
• Alternate Interior Angle property is generally used.

Height (ऊँचाई)

Definition

The vertical distance from the ground to the top of an object is called its Height.

भूमि से किसी वस्तु के शीर्ष तक की लंबवत दूरी उसकी ऊँचाई कहलाती है।

Important Points

• Height is always vertical.
• Represented by h.
• Measured in metre or centimetre.
• Height is generally the perpendicular side.
• Height is calculated using tan θ or sin θ.

Distance (दूरी)

Definition

The horizontal distance between the observer and the object is called Distance.

दर्शक और वस्तु के आधार के बीच की क्षैतिज दूरी को दूरी कहते हैं।

Important Points

• Distance is always horizontal.
• Represented by d.
• Usually forms the base of right triangle.
• Calculated using tan, cos or Pythagoras theorem.
• Unit is metre or kilometre.

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Assumptions Used in NCERT (NCERT में प्रयुक्त मान्यताएँ)

• Surface of Earth is assumed to be flat.
• Objects are considered vertical.
• Observer’s eye and object lie in the same vertical plane.
• Air resistance and Earth’s curvature are ignored.
• Questions are based on right-angled triangles only.
• Measurements are considered accurate.

Trigonometric Ratios Used (प्रयुक्त त्रिकोणमितीय अनुपात)

Important Formulae

$$\tan\theta=\frac{Height}{Distance}$$$$Height=Distance\times\tan\theta$$$$Distance=\frac{Height}{\tan\theta}$$$$\sin\theta=\frac{Perpendicular}{Hypotenuse}$$$$\cos\theta=\frac{Base}{Hypotenuse}$$

Important Points

• tan θ is most frequently used.
• sin and cos are used when hypotenuse is given.
• Choose the ratio according to the given information.
• Always write formula before calculation.

General Steps to Solve Questions (प्रश्न हल करने की सामान्य विधि)

• Read the question carefully.
• Draw a neat labelled diagram.
• Identify height, distance and angle.
• Form a right triangle.
• Choose suitable trigonometric ratio.
• Substitute given values.
• Solve step by step.
• Write final answer with correct unit.
• Check if the answer is reasonable.

Important Standard Values (मानक त्रिकोणमितीय मान)

Angle	sinθ	cosθ	tanθ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	Not Defined

Real Life Applications (वास्तविक जीवन में उपयोग)

• Measuring height of buildings.
• Finding height of mountains.
• Calculating width of rivers.
• Surveying land.
• Designing bridges.
• Aviation and navigation.
• Satellite communication.
• Architecture and construction.
• Military applications.
• Astronomy observations.

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Common Mistakes to Avoid (सामान्य गलतियाँ)

• Confusing angle of elevation with depression.
• Drawing incorrect diagram.
• Using wrong trigonometric ratio.
• Forgetting units.
• Ignoring observer’s eye level.
• Not writing formula.
• Wrong standard values.
• Calculation mistakes.
• Forgetting to simplify surds.

Exam Tips (Board Exam Tips)

• Draw neat diagrams.
• Learn all standard trigonometric values.
• Revise formulas daily.
• Identify opposite and adjacent sides carefully.
• Practice NCERT examples.
• Always mention units.
• Write formula before solving.
• Solve stepwise for full marks.
• Practice previous year questions.
• Keep calculations neat and accurate.

Chapter Summary (अध्याय सारांश)

• Trigonometry helps calculate inaccessible heights and distances.
• Line of Sight joins observer and object.
• Horizontal Line is parallel to the ground.
• Angle of Elevation is measured upward.
• Angle of Depression is measured downward.
• Height is vertical distance.
• Distance is horizontal distance.
• tan θ = Height / Distance is the most useful formula.
• Draw diagrams carefully before solving.
• This chapter is highly scoring and application-based in board examinations.

Frequently Asked Questions (FAQs)