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, "Some Applications of Trigonometry," Class 10 students will explore how trigonometry can be applied to calculate the height and distance of various objects without direct measurement. Historically, astronomers used trigonometry to calculate the distances between planets, Earth, and stars. In modern times, it is commonly used in geography and navigation to determine the latitude and longitude of specific locations.
This chapter will cover key concepts such as heights and distances, the line of sight, the angle of elevation, the horizontal line, and the angle of depression.
Horizontal Level
The horizontal level is the horizontal line through the eye of the observer.
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 is AB is the perpendicular, AC is the hypotenuse, and BC is the base.
Then,
- 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
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
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FAQs on Important Definitions & Formulas: Some Applications of Trigonometry - Mathematics (Maths) Class 10
| 1. What are the basic trigonometric ratios used in height and distance problems? |  |
Ans. The basic trigonometric ratios used in height and distance problems are sine (sin), cosine (cos), and tangent (tan). Specifically, for a right triangle, these ratios are defined as follows:
- Sine of an angle (θ) = Opposite side / Hypotenuse
- Cosine of an angle (θ) = Adjacent side / Hypotenuse
- Tangent of an angle (θ) = Opposite side / Adjacent side.
| 2. How can trigonometric ratios help in calculating the height of a building? | |
Ans. Trigonometric ratios can be used to calculate the height of a building by measuring the distance from a point on the ground to the base of the building and the angle of elevation from that point to the top of the building. By using the tangent ratio, if the distance from the building is 'd' and the angle of elevation is 'θ', the height 'h' can be calculated as:
h = d * tan(θ).
| 3. What is the formula for finding the distance to an object using trigonometry? | |
Ans. To find the distance to an object using trigonometry, you can use the cosine or sine ratios, depending on the information available. For example, if you know the height of the object (h) and the angle of elevation (θ), the distance (d) from the object can be calculated using the formula:
d = h / tan(θ) (using tangent) or
d = h / sin(θ) if the hypotenuse can be measured.
| 4. Can you explain the concept of angle of elevation and its significance in trigonometry? | |
Ans. The angle of elevation is the angle formed between the horizontal line from the observer's eye to the line of sight to an object above the horizontal line. It is significant in trigonometry as it helps in determining the height of objects when the distance from them is known. It is used in various applications, such as surveying and navigation.
| 5. What are some real-life applications of height and distance formulas in trigonometry? | |
Ans. Real-life applications of height and distance formulas in trigonometry include:
- Architecture: Calculating the height of buildings and structures.
- Astronomy: Determining distances to celestial bodies.
- Navigation: Finding the height of mountains or landmarks.
- Construction: Ensuring proper angles and measurements for building projects.
- Surveying: Accurately measuring land and property dimensions.
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