Some Applications of Trigonometry
Some Applications of Trigonometry — Study Notes
NCERT-aligned · 11 notes · 3 shown free
9.1 Heights and Distances
Concept9.1 Heights and Distances
This section introduces the practical application of trigonometric ratios to solve real-life problems involving heights and distances. The fundamental concepts of line of sight, angle of elevation, and angle of depression are explained with the help of diagrams. The line of sight is defined as the line drawn from the eye of an observer to the point in the object viewed. When the object is above the horizontal level, the angle formed between the horizontal and the line of sight is called the angle of elevation. Conversely, when the object is below the horizontal level, the angle formed is called the angle of depression. These angles are crucial in forming right-angled triangles, which allow the use of trigonometric ratios such as sine, cosine, and tangent to calculate unknown heights or distances without direct measurement. The section also discusses the approach to find the height of an object like a minar by knowing the distance from the object, the angle of elevation, and the observer's height. Using the right triangle formed, the tangent ratio is applied to find the height of the object above the observer's eye level, which is then added to the observer's height to get the total height.
- Line of sight is the line from observer's eye to the object viewed.
- Angle of elevation is the angle between horizontal and line of sight when object is above.
- Angle of depression is the angle between horizontal and line of sight when object is below.
- Heights and distances can be calculated using trigonometric ratios in right triangles.
- Observer's height, distance from object, and angle of elevation/depression are key data.
- Tangent ratio is often used to relate height and distance in these problems.
- 📌 Line of sight: Line from observer's eye to the point viewed on the object.
- 📌 Angle of elevation: Angle between horizontal line and line of sight when looking up.
- 📌 Angle of depression: Angle between horizontal line and line of sight when looking down.
Angle of Elevation and Angle of Depression
DefinitionAngle of Elevation and Angle of Depression
This section clearly defines the angle of elevation and angle of depression with the help of diagrams. The angle of elevation is the angle formed by the line of sight with the horizontal when the observer looks at an object above the horizontal level, such as looking up at the top of a minar. The angle of depression is the angle formed by the line of sight with the horizontal when the observer looks at an object below the horizontal level, such as looking down at a flower pot from a balcony. These angles are measured from the horizontal line, upwards for elevation and downwards for depression. The section emphasizes that these angles are equal when observed from the same horizontal line due to alternate interior angles formed by a transversal cutting parallel lines. Understanding these angles is essential for applying trigonometric ratios to solve problems involving heights and distances.
- Angle of elevation is measured upwards from the horizontal to the line of sight.
- Angle of depression is measured downwards from the horizontal to the line of sight.
- Both angles are formed with respect to the horizontal line through the observer's eye.
- Angles of elevation and depression are equal when viewed from the same horizontal level.
- These angles help in forming right triangles for trigonometric calculations.
- 📌 Angle of elevation: Angle between horizontal and line of sight looking upwards.
- 📌 Angle of depression: Angle between horizontal and line of sight looking downwards.
Finding Heights Using Trigonometric Ratios
ExplanationFinding Heights Using Trigonometric Ratios
This section explains the method to find the height of an object such as a minar using trigonometric ratios. Given the distance from the object, the angle of elevation, and the observer's height, the height of the object can be calculated without dir
Practice Questions — Some Applications of Trigonometry
Includes NCERT exercise questions with answers
Q1.In ∆ PQR and ∆ XYZ, ∠ Q = ∠ Y and ∠ R = ∠ Z. Which of the following statements are true? 1. The triangles are congruent and similar 2. The triangles are not congruent and not similar 3. The triangles are similar but cannot say about congruency. 4. None of the above
Answer:
3
Explanation:
[{"id": "d48cebbc-1c07-4634-8147-594a37567c06", "type": "html", "value": " Since two angles are equal, the triangles are similar through AA criterion. No condition for sides are given, hence we cannot say anything about congruency. "}]
Q2.If 35 is removed from the data : 30,34,35,36,37,38,39,40, then the median increases by _____
Answer:
0.5
Explanation:
[{"id": "955f5a62-695f-460d-86b1-0207adf9ab50", "type": "html", "value": " Number of terms = 8 So , the Median will be the mean of the 4th and 5th term = (36 +37) / 2 = 36.5 If 35 is removed , the data is 30,34,36,37,38,39,40 Number of terms = 7 So , the Median will be the 4th term = 37 Difference = 37 ─ 36.5 = 0.5 Hence the correct answer is option 4. "}]
Q3.Following are the marks obtained by 300 students in a competitive examination. Find the median for the following frequency distribution. Marks(x) No. of Students(f) 10 35 20 60 30 84 40 96 50 25
Answer:
30
Explanation:
[{"id": "31089a6b-0b96-4890-9ab7-ff574af72362", "type": "html", "value": " Marks(x) No. of Students(f) Cum. Freq.(cf) 10 35 35 20 60 95 30 84 179 40 96 275 50 25 300 Total Frequency(n) = 300 N/2 = 150 The 150th student falls in the class of 30 marks. So the Median is 30 marks. Hence the correct answer is option 2. "}]
Q4.The frequency distribution table shows the number of mango trees in an orchard and their yield of mangoes. Find the median for the following frequency distribution. No. of mangoes(x) No. of trees(f) 50 - 100 33 100 - 150 30 150 - 200 90 200 - 250 80 250 - 300 17
Answer:
184 mangoes
Explanation:
[{"id": "c06fad95-fd78-41d0-b46c-2aa4970a79bf", "type": "html", "value": " "}]
Q5.What is the angle of elevation of the Sun when the length of the shadow of a vertical pole is equal to its height?
Answer:
45 0
Q6.The height of a tower is 10m. What is the length of its shadow when Sun’s altitude is 45 0 ?
Answer:
10 m
Q7.The angle of elevation of the top of a tower from a point on the ground, which is 20m away from the foot of the tower is 60 0 . Find the height of the tower.
Answer:
20 3 m
Q8.The ratio of the length of a rod and its shadow is 1: \sqrt3 . The angle of elevation of the sun is
Answer:
30 0
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