## NCERT Class 10 Maths Lab Manual – Making of a Clinometer

**Objective**

To make a mathematical instrument ‘clinometer’ to measure the height of a distant object.

**Prerequisite Knowledge**

- Concept of angle of elevation and depression.
- Properties of right-angled triangle.

**Materials Required**

Small pipe or drinking straw, a wooden board, wooden strip, thread, weight, screw, geometry box, etc.

**Procedure**

- Prepare a semi-circular protractor with the help of geometry box. Mark degrees in sexagesimal scale with 0° at the lowest and 10 to 90° proceeding both clockwise and anti-clockwise.
- Fix a hollow pipe along the diameter of it fig. (i).
- Punch a whole at the centre of a semicircle.
- Suspend a weight (w) from a small nail fixed to the centre.
- Ensure that the weight at the end of the string hangs below the protractor.

** Determining the Height of an Object:**

6. Measure the distance of the object from you. Let it is d.

7. Look through the hollow pipe straw at the top of the object by rotating it gradually. Make sure that you can clearly see the top of the object.

8. Hold the clinometer steady and record the angle which the string makes on the scale of the clinometer. This angle is the required angle of elevation , let be θ .

Using trigonometric ratio:

tan θ = \(\frac { height }{ distance } \) =\(\quad \frac { h }{ d } \)

h=d.tan θ

**Observation**

- angle of elevation 0.
- distance between object and clinometer d.
- height of clinometer/(let)

Therefore height of object =l + h = l+dtanθ

**Result**

Angle of elevation can be found easily.

**Learning Outcome**

Students learn how to determine the angle of elevation of an object by clinometer and use of it to determine the height of an object at a known distance.

**Activity Time**

- If the height of clinometer is 3 m, distance between object and clinometer is 27 m and angle of elevation is 45°. What is the height of an object?
- A kite is flying at a height of \(40\sqrt { 3 } \) metre from the ground level, attached to a string inclined at 60° to the horizontal. What is the length of the string ?

You can also download **NCERT Maths Class 10** to help you to revise complete syllabus and score more marks in your examinations.

**Viva Voce **

**Question 1:**

What is trigonometry

**Answer:**

It is an important branch of mathematics. In this branch we deal with the relation and measurement of the sides and the angles of a triangle.

**Question 2:**.

How many trigonometric ratios are there for an acute angle in a right angled triangle? .

**Answer:**

6.

**Question 3:**

Is each trigonometric ratip a real number ?

**Answer:**

Yes.

**Question 4:**

Which triangle is used in trigonometry?

**Answer:**

Right-angled triangle

**Question 5:**

If the angle of elevation of an object is 30°, then tan 0 (where 0 is angle of elevation) is

**Answer:**

tan θ = tan 30°= \(\frac { 1 }{\sqrt { 3 }}\)

**Question 6:**

In AABC, right angled at B, AB = 24 cm and BC = 7 cm. Determine sin A, cos A, sin C and cos C.

**Answer:**

sin A =\(\frac { 7 }{ 25 } \) , cos A =\(\frac { 24 }{ 25 } \) , sin C =\(\frac { 24 }{ 25 } \) , cos C =\(\frac { 7 }{ 25 } \)

**Question 7:**

If tan A = cot B, then is it right A + B = 90° ?

**Answer:**

Yes

**Question 8:**

What is the value of \(\frac { sin\quad { 18 }^{ 0 } }{ cos\quad { 72 }^{ 0 } } \)?

**Answer:**

1

**Multiple Choice Questions**

**Question 1:**

If 3 cot A = 4, then \(\frac { 1-{ tan }^{ 2 }A }{ 1+{ tan }^{ 2 }A }\) is equal to

(a) \({ cos }^{ 2 }A-{ sin }^{ 2 }A\)

(b) \({ sin }^{ 2 }A-{ cos }^{ 2 }A\)

(c) \({ sin }^{ 2 }A+{ cos }^{ 2 }A\)

(d) none of these

**Question 2:**

In ΔABC, right angled at B, if tan A =\(\frac { 1 }{\sqrt { 3 } }\), then the value of sin A cos C + cos A sin C is

(a) 0

(b) 1

(c) 2

(d) none of these

**Question 3:**.

In ΔABC right angled at C if ∠ A = ∠B, then which is correct ?

(a) cos A = cos B

(b) cos A = cos C

(c) cos B = cos C

(d) none of these

**Question 4:**

Choose the correct option for, \(\frac { { 2\quad tan\quad 30 }^{ 0 } }{ 1+{ tan }^{ 2 }30 } \) =

(a) sin 60°

(b) cos 60°

(c) tan 60°

(d) sin 30°

**Question 5:**

If sin (A + B) = 1, cos (A – B) = 1, 0° < A + B ≤90°, A ≥ B, then A and B are respectively

(a) 45° and 45°

(b) 45° and 15°

(c) 30° and 30°

(d) 30° and 15°

**Question 6:**

Choose the correct option for\(\quad 9{ sec }^{ 2 }-9{ tan }^{ 2 }\)

(a) 1

(b) 9

(c) 8

(d) 0

**Question 7:**

Choose the correct option for (sec A) (1 — sin A) (sec A + tan A).

(a) 1

(b) 2

(c) 0

(d) -1

**Question 8:
**A player sitting on the top of a tower of height 20 m observes the angle of depression of a ball tied on the ground as 60°. What is the distance between the foot of the tower and the ball ?

(a) 11.45 m

(b) 11.55 m

(c) 11.59 m

(d) none of these

**Question 9:
**A plane is observed to be approaching the airport. It is at a distance of 12 km from the point of observation and makes an angle of elevation of 30° at that point. What is its height above the ground ?(a) 5 km

(b) 4 km

(c) 6 km

(d) 7 km

**Question 10:**

If tan (60° —α) = 1, then α is

(a) 45°

(b) 15°

(c) 60°

(d) 30°

**Question 11:**

The value of 3 cos2 30° + tan 60° is

(a) \(\frac { 21 }{ 4 } \)

(b) 5

(c) 1

(d) 6\(\frac { 21 }{ 4 } \)

**Answers
**1. (a)

2. (b)

3. (a)

4. (a)

5. (a)

6. (b)

7. (a)

8. (b)

9. (c)

10. (b)

11. (a)

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