‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

Define the angle of elevation.

The angle formed by the line of sight with the horizontal when the object viewed is above the horizontal level.

Which trigonometric ratio links the opposite side and the hypotenuse?

The sine ratio, \(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\).

What is the line of sight?

The straight line drawn from the eye of an observer to the point being viewed on the object.

Define the angle of elevation.

The angle between the horizontal and the line of sight when the object viewed is above the horizontal level (refer to the figure in your NCERT textbook).

Define the angle of depression.

The angle between the horizontal and the line of sight when the object viewed is below the horizontal level.

Which trigonometric ratio links the height and the base in a height-and-distance problem?

The tangent ratio, since \(\tan\theta = \dfrac{\text{height (opposite)}}{\text{horizontal distance (adjacent)}}\).

A tower casts a shadow such that the angle of elevation of the sun is \(45^\circ\). How are the height and shadow related?

Since \(\tan 45^\circ = 1 = \dfrac{\text{height}}{\text{shadow}}\), the height of the tower equals the length of its shadow.

The angle of elevation of the top of a 30 m tower from a point on the ground is \(30^\circ\). Find the distance of the point from the base.

\(\tan 30^\circ = \dfrac{30}{d} \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{30}{d} \Rightarrow d = 30\sqrt{3}\) m.

Why does the angle of elevation increase as one moves towards a tower?

Because the horizontal distance decreases while the height stays the same, so the line of sight makes a larger angle with the horizontal.

What practical problems can be solved using trigonometry?

Finding the heights of towers, buildings, hills and trees, and the widths of rivers or distances of ships, without measuring them directly.

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Chapter 9: Some Applications of TrigonometryClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

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The angle of elevation is the angle made by the line of sight with the horizontal when the object is:

Summary

This chapter applies trigonometric ratios to practical problems of heights and distances that cannot be measured directly. The key ideas are the angle of elevation (looking up at an object) and the angle of depression (looking down). By modelling a situation as a right-angled triangle and using the appropriate ratio, students can find the height of a building, the width of a river or the distance of a ship. The chapter strengthens the ability to translate a real scene into a diagram and solve it with trigonometry.

Angle of elevationAngle of depressionHeights and distances problems

Key terms

Angle of elevation

The angle above the horizontal when looking up at an object.

Angle of depression

The angle below the horizontal when looking down at an object.

Line of sight

The straight line from the observer’s eye to the object.

Heights and distances

Problems solved using right-triangle trigonometry.

Extra questions & answers

EXERCISE 9.1

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Height from angle of elevation

Angle of elevation (°)Distance from base (m)

Height =11.55m

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Angle of elevation

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Meaning1 / 4

The angle above the horizontal when looking up at an object.

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