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Class 10 Maths Revision Notes for Some Applications of Trigonometry of Chapter 9 - Free PDF Download
Some Applications of Trigonometry Class 10 Notes CBSE Maths Chapter 9 (Free PDF Download)
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Angles of Elevation and Depression:
The line of sight is a line drawn from an observer's eye to a point in the thing being observed.
Angle of Elevation:
When the point being observed is above the horizontal level, the angle created by the line of sight with the horizontal is called the angle of elevation of the point being viewed.
Let $\text{O}$ and $\text{P}$ be two points, with $\text{P}$ being the greater level. Assume that $\text{OA}$ and $\text{PB}$ are horizontal lines that pass through $\text{O}$ and $\text{P}$, respectively.
If the observer is at $\text{O}$ and the object is at $\text{P}$, the line $\text{OP}$ is known as the line of sight of Point $\text{P}$, and the angle $\text{AOP}$ between the line of sight and the horizontal line $OA$ is known as the angle of elevation of Point $\text{P}$ as seen from $\text{O}$.
Angle of Depression:
The angle created by the line of sight with the horizontal when the point is below the horizontal level is known as the angle of depression of a point on the object being viewed.
The angle $\text{BPO}$ is known as the angle of depression of $\text{O}$ as seen from $\text{P}$ if the observer is at $\text{P}$ and the object under examination is at $\text{O}$.
From the diagram, it is obviously clear that the angle of elevation of a point $\text{P}$when viewed from a point $\text{O}$ is equal to the angle of depression of $\text{O}$ when viewed from $\text{P}$.
Example:
1. From the top of a $\text{7 m}$ high building, the angle of elevation of the top of a cable tower is \[\text{6}{{\text{0}}^{\text{0}}}\] and the angle of depression of its foot is \[{{45}^{\text{0}}}\] . Determine the height of the tower.
Ans:
Given Data:
Given the height of building $\text{= 7 m}$
Angle of the elevation of top of a tower from the top of building \[\text{= 6}{{\text{0}}^{\text{0}}}\]
Angle of the depression of the foot of a tower \[\text{= 4}{{\text{5}}^{\text{0}}}\]
To find:
We need to calculate the height of the tower, that is \[\text{AE}\]
Sol:
From the given data, we have
Angle of elevation, that is
\[\angle \text{ACB = 6}{{\text{0}}^{\text{0}}}\]
Angle of depression, that is
\[\angle \text{BCE = 4}{{\text{5}}^{\text{0}}}\]
Here,
$BE = CD = 7 m$
In the right angled triangle \[ABC\] ,
\[\text{tan }\!\!\theta\!\!\text{ = }\dfrac{\text{Opposite Side}}{\text{Adjacent Side}}\]
Therefore,
$\text{tan 6}{{\text{0}}^{0}}\text{ = }\dfrac{\text{Opposite Side}}{\text{Adjacent Side}}$
$\text{tan 6}{{\text{0}}^{0}}\text{ = }\dfrac{\text{h}}{\text{CB}}$
Thus,
\[\text{h = CB }\!\!\times\!\!\text{ tan 6}{{\text{0}}^{\text{0}}}\]
Since,
\[\text{tan 6}{{\text{0}}^{\text{0}}}=\sqrt{3}\]
Therefore,
$\text{h = CB }\!\!\times\!\!\text{ }\sqrt{\text{3}}$
$\text{h = }\sqrt{\text{3}}\text{ CB ------}\left( \text{1} \right)$
In the right angled triangle \[CBE\] ,
\[\text{tan }\!\!\theta\!\!\text{ = }\dfrac{\text{Opposite Side}}{\text{Adjacent Side}}\]
Therefore,
$\text{tan 4}{{\text{5}}^{0}}\text{ = }\dfrac{\text{Opposite Side}}{\text{Adjacent Side}}$
$\text{tan 4}{{\text{5}}^{0}}\text{ = }\dfrac{7}{\text{CB}}$
Since,
\[\text{tan 4}{{\text{5}}^{\text{0}}}=1\]
Therefore,
$\text{ 7 = CB }\!\!\times\!\!\text{ 1}$
$\text{CB = 7 ------}\left( \text{2} \right)$
By putting the equation \[\left( 2 \right)\] in the equation \[\left( 1 \right)\], we get
$\text{ h = 7 }\!\!\times\!\!\text{ }\sqrt{\text{3}}$
$\text{AB = 7}\sqrt{\text{3}}$
Therefore, total height of the tower can be calculated as,
Total height of the tower,
$\text{AE = AB + BE}$
$\text{AE = 7}\sqrt{3}+7$
$\text{AE = 7}\left( \sqrt{3}+1 \right)$
Therefore, the total height of the tower is \[\text{7}\left( \sqrt{3}+1 \right)\] unit.
Class 10 Maths Chapter 9 Revision Notes
Class 10 Maths Notes of Some Applications of Trigonometry
Trigonometry is a mathematical concept that we apply on angles and sides of triangles and it serves as a very important and scoring topic for the Class 10 syllabus. The chapters and their weightage for Class 10 board exams are mentioned below in the table:
You will learn how to apply trigonometry concepts on triangles, how to deal with trigonometric ratios, angles and sides, angle of elevation, angle of depression, etc. in this chapter. With the help of various trigonometric formulas and concepts, you will be able to solve statement-based questions of this chapter as well. We will cover Class 10 Maths Chapter 9 revision notes in this article and will see the various benefits of these notes:
Benefits of Revision Notes Class 10 Maths Chapter 9
Class 10 revision notes Maths Ch 9 are designed in such a way that students will be able to go through all the concepts at once in very little time and they will not miss any important concept related to the chapter.
Revision notes of Chapter 9 will benefit the students in a number of ways such as it is going to help them in speeding up the problem-solving skills of the students and for this, we suggest practising and revising these notes regularly.
Class 10 Revision Notes Some Applications of Trigonometry will help you in covering the important and scoring topics of the syllabus that is Trigonometry and Chapter 9 is considered as one of the toughest and scoring chapters and these notes will help you in preparing this topic.
The Class 10 Revision Notes Maths Ch 9 created and provided by Vedantu are in a very illustrative and descriptive way. While preparing these notes, we have kept every important point in our mind like your exam pattern, important concepts which need to be highlighted, again and again, all the concepts related to Chapter 9 and Trigonometry, etc.
With Maths Class 10 Some Applications of Trigonometry Notes, Students will be able to learn and revise all the formulas or concepts even a moment before any exam and this last moment revision will be very beneficial also because it will accelerate their exam preparation and will help you in solving long statement based questions.
Notes of Class 10 revision Notes Chapter 9 will be beneficial for you if students build the habit of revising these notes always before they start with problem-solving. The revision notes are designed by experts by keeping in mind the students' need for last-minute revision before attempting any chapter so they can optimize their results.
Before any test or exam, you will not have to turn all the pages of the Trigonometry chapter in a book to revise the concepts. Some Applications of Trigonometry Class 10 notes will solve this problem and you will not miss any important point at the end while revisioning.
Preparing notes for revision at the end in such a way you take less time in covering any chapter is very time-consuming but important as well and these Class 10 Maths Revision Notes Chapter 9 will help you to save that time and that time you can use actually to practice which is more important to master the concepts.
General Tips
While revising, go through every point and concept related to the chapter carefully in Ch 9 Class 10 Maths revision notes because it is one of the toughest chapters of Class 10.
Prefer revising it every time before you start solving the problems of the chapters because in these chapters you have to deal with statement-based questions and need to make their diagrams as well, only then you will be able to solve the question at the end.
Do also revise the notes of Chapter 8 related to Introduction to Trigonometry because some of the basic concepts of that chapter will also be used in Chapter 9.
You can revise it again in future grades before you start with the Trigonometry chapter in upper classes as well.
Solved Questions
1. The angle of elevation of the top of a tower from a point on the ground, which is at a distance of 30 m from the foot of the tower, is 30°. Determine the height of the tower.
Solution:
Let AB be the height of the tower and C be the point of elevation, which is at a distance of 30 m from the foot of the tower.
(Image Will Be Updated Soon)
In right ⊿ ABC
tan 30° = AB/BC
1/√3 = AB/30
⇒ AB = 10√3
Therefore, the height of the tower is 10√3 m.
2. A kite flies at a height of 60 m above the level of the ground. Its string is temporarily tied to a point on the ground. The string’s inclination with the ground is at 60°. Determine the length of this string, considering that the string doesn’t have any slack.
Solution:
Sketch out a figure based on the given information.
(Image Will Be Updated Soon)
From the figure BC = Height of the kite from the ground = 60 m, AC = Inclined length of the string from the ground, and A be the point where the string of the kite is tied.
From the drawn figure,
sin 60° = BC/AC
⇒ √3/2 = 60/AC
⇒ AC = 40√3 m
Therefore, the length of the string from the ground is 40√3 m.
Other Maths Related Links
CBSE Class 10 Revision Notes - Other Chapters
The following list consists of links that drive you to the revision notes for each chapter in the CBSE Class 10 syllabus. We recommend that students check these links out to make sure that they don’t miss out on the expert guidance that our experts provide for improving their scores in exams.
Benefits of Revision Notes Class 10 Maths Chapter 9
Comprehensive Concept Review:
Designed for a quick yet comprehensive review of all concepts in Chapter 9.
Ensures no crucial topic related to the chapter is missed during revision.
Enhanced Problem-Solving Skills:
Benefits students by accelerating problem-solving skills.
Regular practice and revision of these notes are recommended for optimal results.
Coverage of Important and Scoring Topics:
Focuses on crucial and scoring topics of the syllabus, particularly Trigonometry.
Assists in preparing for Chapter 9, considered both challenging and rewarding.
Illustrative and Descriptive Presentation:
Vedantu's Revision Notes are illustrative and descriptive, considering exam patterns and highlighting important concepts.
Emphasizes key points related to Chapter 9 and Trigonometry for effective learning.
Last-Minute Revision Benefits:
Enables last-minute revision with the ability to learn and revise formulas or concepts just before exams.
Particularly beneficial for solving lengthy statement-based questions.
Effective Habits for Success:
Encourages the habit of revising notes before problem-solving sessions.
Expertly designed for last-minute revision, optimizing results for students.
Convenient Revision Process:
Eliminates the need to flip through entire Trigonometry chapters before tests.
Solves the problem of missing crucial points during revision.
Time-Efficient Preparation:
Saves time in preparing comprehensive revision notes for Chapter 9.
Allows students to allocate saved time for essential practice sessions, crucial for mastering concepts.
Incorporated in the CBSE Class 10 Maths syllabus, this Trigonometry chapter emphasizes a practical approach, aiding students in mastering its application in daily life and exams. To facilitate easy comprehension, our expert teachers have curated study materials that simplify the topic without unnecessary complexity. We recommend students delve into these revision notes and explore related links in this article to optimize their learning experience and make the most of the valuable insights provided by our experts.
Other Maths Related Links
Conclusion:
When it comes to Class 10 Maths, Trigonometry stands out as a crucial and high-scoring topic, comprising two vital chapters. Questions from this area are a recurring feature in exams, emphasizing the need for thorough practice. Vedantu addresses this with Class 10 Maths Notes on Some Applications of Trigonometry, aiding in last-minute revision. Tailored to the syllabus and board exam requirements, these notes save time, allowing students to concentrate on studies. Designed for those aiming for good marks, Vedantu's notes ensure a quick recap without missing any details, empowering students to confidently solve problems from this chapter.
FAQs on Some Applications of Trigonometry Class 10 Notes CBSE Maths Chapter 9 (Free PDF Download)
1. How can I Boost my Maths Preparation for Class 10 Board Exams?
Maths exam of Class 10 is a standardized paper with a standardized syllabus that helps you in building good subject knowledge with different topics and you can boost your preparation with NCERTs and other standard books, Class 10 Maths Chapter 9 Revision notes provided by Vedantu, sample papers or mocks tests and previous year papers, etc. All these things will help you in getting good marks in this subject in board exams.
2. How to Make Revision Notes for Class 10 Maths?
Undoubtedly, making revision notes on our own is not easy and is a very time-consuming task but one cannot ignore its importance and value in board exams. Thus, Vedantu has taken up the initiative by providing the revision notes of all the chapters of Maths of Class 10 separately so that students can utilize these notes along with their preparation and can get good marks. You can download these notes of any chapter from the website of Vedantu and start utilizing them during your preparation.
3. What is the Major Benefit of Revision Notes Class 10 Maths Chapter 9?
Revision notes of any chapter or topic or any syllabus are very important to revise the concept in less time and Chapter 9 of Class 10 Maths is one of the most important and toughest chapters of Class 10 that needs a lot of practice and revision and these notes revision notes Class 10 Maths Chapter 9 will help you in covering this chapter that will help you in solving questions of this chapter.
4. What is the line of sight and angle of elevation?
The line of sight is the imaginary line that is drawn from the eye of an observer to the point in an object that is viewed by the observer. The angle of elevation is the angle that is made between the line of sight and the horizontal line, where the object that we see lies above this horizontal line, in the sense, we have to raise our head above the horizontal line to look at the point in an object.
5. What is the angle of depression?
The angle of depression is the angle that is produced when the line of sight and the horizontal meet, where the object that is viewed by us is well below the horizontal line. So, in the case of the angle of depression, we are looking downwards at a point in an object. A good example to explain the angle of expression could be a person looking down at something from their balcony.
6. Where can I get the best notes for Chapter 9 Class 10 Maths?
The best notes for students to study Class 10 Maths Chapter 9 is Vedantu’s CBSE Class 10 Maths Notes Chapter 9 Some Applications of Trigonometry. The reason why these are the best notes for you is because of their comprehensiveness. All the important concepts like angle of elevation, angle of depression, line of sight etc have been explained in a simplistic way for students to grasp. You will find multiple examples that will help in clearing your basics completely.
7. Are all the exercises essential for Chapter 9 Class 10 Maths?
Yes, all exercises present in NCERT Class 10 Maths Chapter 09 “Some Applications of Trigonometry” are extremely important from an examination point of view. If you are getting stuck in exercises, you can always rely on Vedantu’s NCERT Solution for Class 10 Maths to get expert answers solved step by step. To get notes from the chapter to strengthen your core ideas and concepts, you can use CBSE Class 10 Maths Notes Chapter 9 Some Applications of Trigonometry at Vedantu.
8. How do you calculate the height and distance between an object and an observer?
If you want to calculate the height of an object or the distance between two objects at an angle of depression or an angle of elevation, you will have to use trigonometric ratios, especially tan A or cot A. If you want access to more important points, refer to CBSE Class 10 Maths Notes Chapter 9 Some Applications of Trigonometry compiled by experts for students. They have made student-friendly notes that cover all the points required to complete the chapter properly.
