Class 10 Exam > Class 10 Notes > Mathematics (Maths) Class 10 > Important Definitions & Formulas: Some Applications of Trigonometry
Important Definitions & Formulas: Some Applications of Trigonometry | Mathematics (Maths) Class 10 PDF Download
Introduction
In Chapter 9, Class 10 students will learn how to use trigonometry to find the height and distance of objects without measuring them directly. In the past, astronomers relied on trigonometry to calculate distances between planets, Earth, and stars. This chapter explores various ways trigonometry is applied in everyday life.
Topics covered in this chapter include:
- Heights and distances
- The line of sight
- The angle of elevation
- The horizontal line
- The angle of depression
By using trigonometric ratios, we can determine the height or length of an object or the distance between two far-away objects.
Horizontal Level
The horizontal level refers to the straight line that runs through the observer's eye.
Line of Sight
The line which is drawn from the eyes of the observer to the point being viewed on the object is known as the line of sight.
Angle of Elevation
The Angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object.
Angle of Depression
The angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed.
Height and Distance Formulas for Trigonometric ratio
ΔABC is a right-angled triangle where the side AB is the vertical side, AC is the longest side (the hypotenuse), and BC is the horizontal side.
The trigonometric ratios are defined as follows:
- Sin θ = AB/AC
- Cos θ = BC/AC
- Tan θ = AB/BC
- Cosec θ = AC/AB
- Sec θ = AC/BC
- Cot θ = BC/AB
Trigonometrical Identities:
- sin2θ + cos2 θ = 1
- 1 + tan2θ = sec2 θ
- 1 + cot2θ = cosec2 θ
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Some Examples Using Above Formulas
Q1: Adam is standing in the Sun. He is 6ft tall and is casting a shadow of 4ft. A flag is also present near the man casting a shadow of 36ft. Calculate the height of the flag.
Ans: The angle of elevation for the Sun will be the same for both the cases.
Therefore, the ratio of object to shadow will be the same for both the cases as per the proportionality rule.
Object height/ Shadow height = 6/ 4 = H /36
Therefore, H = 54 ft = Height of the flag
Ans: The angle of elevation for the Sun will be the same for both the cases.
Therefore, the ratio of object to shadow will be the same for both the cases as per the proportionality rule.
Object height/ Shadow height = 6/ 4 = H /36
Therefore, H = 54 ft = Height of the flag
Q2: When a boy looks from the foot and the top of a tower at the roof of a building, the angles of elevation and depression are 27o and 63o, The height of this building is 40m, then calculate the height of the tower given that tan 630 =2.
Ans: Let the tower be AB
Let the building of height 40m be CD
In the given triangle ACD, AC/DC = cot 270
= cot (90-63)
AC/40 = tan 630 = 2
AC = 80m
Now, DE = AC = 80m
Also, in triangle BED, tan 630 = BE/DE
2= BE/80
Therefore, BE = 160 m
Therefore, the height of the tower can be calculated as AE + EB
= 40 +160 = 200m
The document Important Definitions & Formulas: Some Applications of Trigonometry | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Important Definitions & Formulas: Some Applications of Trigonometry - Mathematics (Maths) Class 10
| 1. What are the key trigonometric ratios used in height and distance problems? |  |
| 2. How can we calculate the height of a tree using trigonometry? | |
Ans. To calculate the height of a tree using trigonometry, measure the distance from the tree (let's call it \( d \)) and the angle of elevation (\( \theta \)) from the ground to the top of the tree. The height (\( h \)) can be calculated using the formula:
\[ h = d \cdot \tan \theta \]
This formula arises from the tangent ratio, where the height of the tree is the opposite side, and the distance from the tree is the adjacent side.
| 3. What is the formula to find the distance of an object from a point when the angle of depression is known? | |
Ans. When the angle of depression (\( \theta \)) from a point to an object is known, the distance (\( d \)) to the object can be calculated using the formula:
\[ d = h \cdot \tan \theta \]
where \( h \) is the height from which the angle is measured. This is derived from the tangent ratio in the right-angled triangle formed by the height and the distance to the object.
| 4. What role does the angle of elevation play in solving height and distance problems? | |
Ans. The angle of elevation is the angle formed between the horizontal line and the line of sight to an object above the horizontal line. In height and distance problems, it is crucial for determining the height of an object when the distance from the object is known. The relationship can be expressed using the tangent function:
\[ \tan \theta = \frac{\text{Height}}{\text{Distance}} \]
This helps in finding unknown heights by rearranging the formula based on known values.
| 5. Can you explain the relationship between angles of elevation and depression? | |
Ans. The angle of elevation and the angle of depression are related in that they are equal when measured from a horizontal line. The angle of elevation is measured upwards from the horizontal to a point above, while the angle of depression is measured downwards from the horizontal to a point below. In practical problems, if you know the angle of elevation from point A to point B, the angle of depression from point B to point A will be the same, which can be useful in solving height and distance problems.
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