"The Cracker" Practice Book for Trigonometry and Height & Distance

PREFACE Students, before you lies a book, “The Cracker Practice Book for Trigonometry and Height & Distance”, the basis of which, are the fundamental concepts of trigonometry that one needs to have all grasped before she appears for SSC CGL, CPO, CHSL or any other examination. This book has been prepared to fulfil the essential and fundamental requirements of the government job aspirants who will appear for the examinations in which questions based on trigonometry are asked widely. As per a recent change observed in the trend of these examinations, the trigonometry questions now being asked are difficult to solve when compared to previous examinations. Also, advanced mathematics makes fifty to sixty per cent part of the Mains examination of SSC CGL out of which around thirty per cent questions are based on Trigonometry and Height & Distance. So, it becomes a must for all the SSC aspirants not to give this portion a cold shoulder. This book also covers the difficult new pattern questions under the name Challenger Practice Sets along with the previous year questions that are expected to be repeated in the upcoming examinations. To describe and explain the concepts of trigonometry and its applications, this book uses the tricks and formulas that will help the government job aspirants solve the most difficult questions in the minimum time possible. It is projected to implant the approaches in the aspirant's mind to solve all types of trigonometry questions that could be asked in the upcoming SSC CGL, CPO, CHSL examinations for various recruitment processes. In this book, the detailed theories are followed by practice exercises and previous year questions that help the student to analyze what is being asked in these examinations so that she prepares accordingly. We would like to thank experienced faculties and subject-matter experts at Adda247, without whose cooperation, it wouldn't have been possible to bring forward this meticulous study material for SSC CGL, CPO, CHSL and other competitive exams. We hope that our readers appreciate the strenuous efforts that we put into this book. Any remarks or suggestions for further refinements are wholeheartedly welcome. Team Adda247

CONTENTS I.

Important Concepts & Formulae for Trigonometry . . . . . . . . . . . . . . . . . . . 5

II.

Previous Year Questions 1.

SSC CGL Tier II 2017 – 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.

SSC CGL Tier II 2016 – 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.

SSC CGL Tier II 2015 – 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.

SSC CPO 2017 – 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.

SSC CGL Tier I 2017 – 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.

SSC CGL Tier I 2016 – 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

III. 15 Practice Sets Practice Set 01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Practice Set 02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Practice Set 03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Practice Set 04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Practice Set 05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Practice Set 06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Practice Set 07 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Practice Set 08 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Practice Set 09 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Practice Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Practice Set 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Practice Set 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Practice Set 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Practice Set 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Practice Set 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

IV. 10 Challenger Practice Sets Challenger Practice Set 01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Challenger Practice Set 02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Challenger Practice Set 03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Challenger Practice Set 04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Challenger Practice Set 05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Challenger Practice Set 06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Challenger Practice Set 07 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Challenger Practice Set 08 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Challenger Practice Set 09 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Challenger Practice Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

"The Cracker" Practice Book for Trigonometry

Important Concepts & Formulae for Trigonometry Trigonometry: Trigonometry word is derived from two Greek words. • Trigonon: Triangle • Metron : Measure So, generally, it is a branch of mathematics that defines the relationship between the angle and the sides of the triangle. Measurement of angle: In circular system, unit of measurement of angle is radian. It is denoted by 1c. 1 Radian: It is the angle subtended at the centre by the arc of length equal to radius of circle. • In circle, circumference = π × diameter Where π = 3.1416 or

22 7

• •

Relationship between degree and radian: π radian = 180° When an arc subtends an angle θ radian at the centre of a circle of radius r then, arc θ = radius

•

1 radian =

•

Thus, to change degree into radian, multiply by

180 22 7

( )

= 57° 16' 22'' π 180°

and to change radian into degree multiply by

180° π

Example: Find radian measure of 120°. Solution: 120° = 120 ×

π

2

= π radian

180°

3

3

Example: Find the value of 5 radian in degree. 3

3

Solution: 5 rad = 5 ×

180 π

3

7

= 5 × 22 × 180° =

378° 11

= 34° 21' 49''

Trigonometric Ratio:

In right angle triangle DAC, There are six trigonometric ratios.

5

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"The Cracker" Practice Book for Trigonometry

Some important relations 1 • sin θ = → sin θ.cosec θ = 1 cosec θ 1

•

cos θ = sec θ → sec θ.cos θ = 1

•

tan θ =

•

tan θ = 𝑐𝑜𝑠 𝜃

•

cot θ = 𝑠𝑖𝑛 𝜃

1

cot θ 𝑠𝑖𝑛 𝜃

→ tan θ.cot θ = 1

𝑐𝑜𝑠 𝜃

Trigonometric ratio of some specific angles

Some Important Result • If sin x = 0 or tan x = 0, then x = nπ • Sin 15° = cos 75° • sin 22

1° 2 1°

=

√3 − 1 = 2√2

√2 − √2

1° 1°

• tan 22 2 = √2 – 1

√3 + 1 2√2

𝜋 2

√2 + √2 2

• cot 22 2 = √2 + 1 √5 − 1 4 √10 − 2√5

• sin 36° = cos 54° =

6

• cos 15° = sin 75° = • cot 22 2 =

2

• sin 18° = cos 72°=

If cos x = 0 or cot x = 0, then x = (2n + 1)

4

• cos 18° = sin 72° = • cos 36° = sin 54° =

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√10 + 2√5 4 √5 + 1 4

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"The Cracker" Practice Book for Trigonometry

Sign of Trigonometric functions in different Quadrants We can obtain the sign of trigonometric function from Quadrant chart. •

We can remember as

Trigonometric ratio of Negative and Associated angle

From the above table we observe that only trigonometric ratio of (90 ± θ), and (270 ± θ) change and they change as. 𝑐ℎ𝑎𝑛𝑔𝑒

sin ↔

𝑐ℎ𝑎𝑛𝑔𝑒

tan ↔

cos cot

𝑐ℎ𝑎𝑛𝑔𝑒

cosec ↔ sec • Trigonometric ratio of (180±θ) and (360±θ) do not change • Sign convention is used according to Quadrant. Example: Example:

𝑠𝑖𝑛 12° 𝑐𝑜𝑠 78°

=

𝑠𝑖𝑛(90°−78°) 𝑐𝑜𝑠 78°

𝑐𝑜𝑠 78°

= 𝑐𝑜𝑠 78° = 1

sin 2295°

So, sin 2295° = (360° × 6 + 135°) = sin 135° Now sin (135°) = sin (90° + 45°) = cos 45° =

1 √2

Some results when A + B = 90° i.e. sum of angles of acute angle triangle = 90° • sin A = cos B • tan A.tan B = 1 • cot A.cot B = 1

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"The Cracker" Practice Book for Trigonometry

Basic Identities • sin² θ + cos² θ = 1 • 1+ tan² θ = sec² θ ⇒ sec² θ – tan² θ = 1 (secθ + tanθ) (secθ – tanθ) = 1, 1 If sec θ + tan θ = P, then, secθ – tanθ = 𝑃 • 1 + cot² θ = cosec² θ ⇒ cosec² θ – cot² θ = 1 (cosec θ + cotθ) (cosec θ – cot θ ) =1, 1 If cosec θ + cot θ = P, then, cosec θ – cot θ = 𝑃 Advanced trigonometric Identities 1. sin (A + B) = sin A cos B + cos A sin B 2. sin (A – B) = sin A cos B – cos A sin B 3. cos (A + B) = cos A cos B – sin A sin B 4. cos (A – B) = cos A cos B + sin A sin B 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 5. tan (A + B) = 1−𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 𝑡𝑎𝑛 𝐴 − 𝑡𝑎𝑛 𝐵

6. 7. 8. 9. 10. 11. 12. 13.

tan (A – B) = 1 + 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 sin (A + B) sin (A – B) = sin² A – sin² B = cos² B – cos² A cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2A 2sin A cos B = sin (A + B) + sin (A – B) 2 cos A sin B = sin (A + B) – sin (A – B) 2cos A cos B = cos (A + B) + cos (A – B) 2sin A sin B = cos (A – B) – cos (A + B) 𝑐𝑜𝑡 𝐴 𝑐𝑜𝑡 𝐵 −1 cot (A + B) = 𝑐𝑜𝑡 𝐴+𝑐𝑜𝑡 𝐵

14.

cot (A – B) =

15.

sin C + sin D = 2sin

16.

sin C – sin D = 2 cos

17.

cos C + cos D = 2 cos

18.

cos C – cos D = 2 sin

19. 20.

sin 2A = 2sin A cos A = 1 + 𝑡𝑎𝑛2 𝐴 cos2A = cos2A – sin2 A = 1 – 2sin2 A = 2 cos2A – 1

21. 22. 23.

tan 2A = = 1−𝑡𝑎𝑛2 𝐴 1 + 𝑡𝑎𝑛2 𝐴 sin 3A = 3 sin A – 4 sin³ A cos 3A = 4 cos³A – 3 cos A

24.

tan 3A =

25.

sin A = 2 sin ( ) cos ( ) = 2 2 1+𝑡𝑎𝑛2(𝐴/2)

26.

cos A = 𝑐𝑜𝑠 2 ( ) − 𝑠𝑖𝑛2 ( ) 2 2

𝑐𝑜𝑡 𝐴 𝑐𝑜𝑡 𝐵 + 1 𝑐𝑜𝑡 𝐴 – 𝑐𝑜𝑡 𝐵 𝐶+𝐷

2 𝑡𝑎𝑛 𝐴

sin

2 𝐶+𝐷 2 𝐶+𝐷 2

𝐶−𝐷

cos

2 𝐶+𝐷

2 𝐶−𝐷

cos sin

2 𝐶−𝐷

2 𝐷−𝐶

2 𝑡𝑎𝑛 𝐴

2

1 − 𝑡𝑎𝑛2 𝐴

3 𝑡𝑎𝑛 𝐴 − 𝑡𝑎𝑛3 𝐴 1−3 𝑡𝑎𝑛2 𝐴 𝐴 𝐴

2 𝑡𝑎𝑛(𝐴/2 )

𝐴

𝐴

= 1 – 2 sin² A/2 = 2 cos² A/2 – 1

2 𝑡𝑎𝑛(𝐴/2) = 1−𝑡𝑎𝑛2(𝐴/2) 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 + 𝑡𝑎𝑛 𝐶 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 𝑡𝑎𝑛 𝐶

27.

tan A

28.

tan (A + B + C) = 1 − 𝑡𝑎𝑛 𝐴.𝑡𝑎𝑛 𝐵 − 𝑡𝑎𝑛 𝐵.𝑡𝑎𝑛 𝐶 − 𝑡𝑎𝑛 𝐴.𝑡𝑎𝑛 𝐶 If A+B+C = 180° • 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 + 𝑡𝑎𝑛 𝐶 = 𝑡𝑎𝑛 𝐴. 𝑡𝑎𝑛 𝐵. 𝑡𝑎𝑛 𝐶 1 1 1 • + 𝑡𝑎𝑛𝐵.𝑡𝑎𝑛𝐶 + 𝑡𝑎𝑛𝐶.𝑡𝑎𝑛𝐴 = 1 𝑡𝑎𝑛𝐴.𝑡𝑎𝑛𝐵 •

8

𝑐𝑜𝑡𝐴. 𝑐𝑜𝑡𝐵 + 𝑐𝑜𝑡𝐵. 𝑐𝑜𝑡𝐶 + 𝑐𝑜𝑡𝐶. 𝑐𝑜𝑡𝐴 = 1

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"The Cracker" Practice Book for Trigonometry 1

29.

sin A sin 2A.sin 4A = sin 3A

30.

cos A cos 2A.cos 4A = cos 3A

31.

tan A tan 2A tan 4A = tan 3A

32.

sin A sin (60-A)sin (60+A) = sin 3A

33.

cos A cos (60-A)sin(60+A)= cos 3A

34.

tan A tan (60-A) tan (60+A)= tan3A

4 1 4

1

4 1 4

Range of trigonometric functions

sin2A and cos2A has minimum value 0 and maximum value 1 so range is (0, 1) For values of 0° ≤ θ ≤ 90° then minimum and maximum value are given as

Sine Rule: In any ∆ABC 𝑎 𝑠𝑖𝑛 𝐴

𝑏

𝑐

= 𝑠𝑖𝑛 𝐵 = 𝑠𝑖𝑛 𝐶 = 2R, where R is circumradius.

9

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"The Cracker" Practice Book for Trigonometry

Cosine Rule: a2 = b2 + c2 – 2bc cos A, b2 = a2 + c2 – 2ac cos B, c2 = a2 + b2 – 2ab cos C Important Results: •

sin θ + cos θ = a then, sin θ – cos θ = √2 − 𝑎2

•

a sin θ + b cos θ = p and, a cos θ – b sin θ = q then, a2 + b2 = p2 + q2, q = ±√𝑎2 + 𝑏2 − 𝑝2

•

a sin θ + b cos θ = √𝑎2 + 𝑏2 then, sin θ =

•

If a sec θ – b tan θ = √𝑎2 + 𝑏2 then, sec θ =

𝑎 √𝑎2 + 𝑏2 𝑎

, cos θ =

√𝑎2 – 𝑏2

𝑏 √𝑎2 + 𝑏2 𝑏

, tan θ = 𝑎

√𝑎2 + 𝑏2

𝑏

•

If a cosec θ – b cot θ = √𝑎2 − 𝑏2 then, cosec θ =

•

Some trigonometric function can be solved easily by considering it as algebraic function.

Example:

√𝑎2 – 𝑏2

, cot θ =

√𝑎2 + 𝑏2

sin θ+ cosec θ = 2 1

It can be considered as x + 𝑥 where x = sin θ 1

1

so, x = 1 and 𝑥 = 1, sin θ = 1 and 𝑐𝑜𝑠𝑒𝑐 𝜃 = 1 Height and Distance Height and distance is one of the important applications of trigonometry. It is a technique used to find the distance and height of the object. We can find them with the help of trigonometric ratios.

Line of Sight: The line of sight is the line drawn from the eye of an observer to the object. Angle of Elevation: When the object is above the horizontal level of our eye, we have to turn our head upwards to see an object. Here ∠POQ is the angle of elevation. Angle of Depression: When the object is below the horizontal level of our eye, we have to turn our head downwards to see an object. Here ∠QOR is the angle of depression. Some Important Points: In this chapter we solve all the questions with the help of ratio.

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"The Cracker" Practice Book for Trigonometry

Concept 1:

If the angle of elevation of the top of a tower at two points which are at a distance of ‘a’ and ‘b’ metres from the foot of tower and on the same side of the tower are complementary. Then height of the tower is √𝑎𝑏.

Example:

If the angle of elevation of the top of a tower at two points which are at a distance of ‘9’ and ‘4’ metres from the foot of the tower and on the same side of the tower are complementary. Find the height of the tower? Height of tower = √𝑎𝑏 = √9 × 4 = √36 = 6 cm

Sol. Concept 2:

If a man wishes to find the height of a tower which stands on a horizontal plane. The angle of elevation of top of the tower is 𝜃1. On walking 𝑥 units towards the tower. He find the angle of elevation becomes 𝜃2. Then the height of the tower is 𝑥 tan 𝜃1 tan 𝜃2

ℎ=[ ℎ=

11

tan 𝜃2 −tan 𝜃1 𝑥

]

cot 𝜃1 −cot 𝜃2

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"The Cracker" Practice Book for Trigonometry

Concept 3:

If two poles of equal heights stand on either sides of a road which is ‘d’ units wide. At a point on the road between the poles, the elevation of the top of the poles are 𝜃1 and 𝜃2, then height of the poles is 𝑑 tan 𝜃1 tan 𝜃2

ℎ=[ ℎ=

Concept 4:

tan 𝜃1 +tan 𝜃2 𝑑

]

cot 𝜃1 +cot 𝜃2

From the top and bottom of the building of height ‘h’ units, the angle of elevation of the top of a tower are ’a’ and ‘b’ respectively, then the height of the tower is [

ℎ 𝑡𝑎𝑛 𝛽

tan 𝛽−tan 𝛼

12

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"The Cracker" Practice Book for Trigonometry

Previous Year Questions

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"The Cracker" Practice Book for Trigonometry

1. What is the value of [(sin x + sin y) (sin x – sin y)]/[(cosx + cosy) (cosy – cosx)]? (a) 0 (b) 1 (c) –1 (d) 2 2. What is the value of [(tan 5θ + tan 3θ)/4 cos 4θ (tan 5θ – tan 3θ)]? (a) sin 2θ (b) cos 2θ (c) tan 4θ (d) cot 2θ

10. A Navy captain going away from a lighthouse at the speed of 4[(√3) – 1] m/s. He observes that it takes him 1 minute to change the angle of elevation of the top of the lighthouse from 60°to 45°. What is the height (in metres) of the lighthouse? (a) 240√3 (b) 480[(√3) – 1] (c) 360√3 (d) 280√2

3. What is the value of (4/3) cot2 (π /6) + 3 cos2 (150°) – 4 cosec2 45° + 8 sin (π /2)? (a) 25/4 (b) 1 (c) – 7/2 (d) 13/2

11. What is the value of [(sin 7x – sin 5x)÷(cos 7x + cos 5x)] – [(cos 6x – cos 4x)÷(sin 6x + sin 4x)]? (a) 1 (b) 2 tan x (c) tan 2 x (d) tan (3x/2)

4. What is the value of sin (B – C) cos (A – D) + sin (A – B) cos (C – D) + sin (C – A) cos (B – D)? (a) 3/2 (b) – 3 (c) 1 (d) 0 5.

{[4 cos(90−A) sin3(90+A)]−[4 sin(90+A) cos3 (90−A)]} cos(

(a) 1 (c) 0

180+8A ) 2

=?

(b) – 1 (d) 2

6. What is the value of cos [(180 – θ)/2] cos [(180 – 9θ)/2] + sin [(180 – 3θ)/2] sin [(180 – 13θ)/2]? (a) sin 2θ sin 4θ (b) cos 2θ cos 6θ (c) sin 2θ sin 6θ (d) cos 2θ cos 4θ 7. What is the value of [tan2(90 – θ) – sin2(90 – θ)] cosec2(90 – θ) cot2 (90 – θ)? (a) 0 (b) 1 (c) – 1 (d) 2 8. Two points P and Q are at the distance of x and y (where y > x) respectively from the base of a building and on a straight line. If the angles of elevation of the top of the building from points P and Q are complementary, then what is the height of the building? (a) xy (b) √y/x (c) √x/y (d) √xy 9. The tops of two poles of height 60 metres and 35 metres are connected by a rope. If the rope makes an angle with the horizontal whose tangent is 5/9 metres, then what is the distance (in metres) between the two poles? (a) 63 (b) 30 (c) 25 (d) 45 14

12. What is the value of [(cos³ 2θ + 3 cos 2θ)÷(cos⁶ θ – sin⁶ θ)]? (a) 0 (b) 1 (c) 4 (d) 2 π

3π

13. What is the value of tan( 4 + A) × tan ( 4 + A) ? (a) 1 (c) cot A/2

(b) 0 (d) – 1

14. What is the value of [(sec 2θ + 1)√sec 2 θ– 1] × 1 2

(cot θ– tan θ)?

(a) 0 (c) cosec θ

(b) 1 (d) sec θ

15. What is the value of sin (630° + A) + cos A? (a) √3/2 (b) 1/2 (c) 0 (d) 2/√3 16. What is the value of [(sin 59° cos 31° + cos 59° sin 31°)÷(cos 20° cos 25° – sin 20° sin 25°)]? (a) 1/√2 (b) 2√2 (c) √3 (d) √2 17. What is the value of cos (90 – B) sin (C – A) + sin (90 + A) cos (B + C) – sin (90 – C) cos (A + B)? (a) 1 (b) sin ( A+ B – C) (c) cos ( B+ C – A) (d) 0 18. Two trees are standing along the opposite sides of a road. Distance between the two trees is 400 metres. There is a point on the road between the trees.The angle of depressions of the point

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"The Cracker" Practice Book for Trigonometry

from the top of the trees are 45° and 60°. If the height of the tree which makes 45° angle is 200 metres, then what will be theheight (in metres) of the other tree? (a) 200 (b) 200√3 (c) 100√3 (d) 250 19. A tower stands on the top of a building which is 40 metres high. The angle of depression of a point situated on the ground from the top and bottom of thetower are found to be 60° and 45° respectively. What is the height (in metres) of tower? (a) 20√3 (b) 30(√3 + 1) (c) 40(√3 – 1) (d) 50(√3 – 1) 20. From a point P, the angle of elevation of a tower is such that its tangent is 3/4. On walking 560 metres towards the tower the tangent of the angle ofelevation of the tower becomes 4/3. What is the height (in metres) of the tower? (a) 720 (b) 960 (c) 840 (d) 1030

27. What is the value of [cos (90 + A)÷sec (270 – A)] + [sin (270 + A)÷cosec (630 – A)]? (a) 3 sec A (b) tan A secA (c) 0 (d) 1 28. On walking 100 metres towards a building in a horizontal line, the angle of elevation of its top changes from 45° to 60°. What will be the height (inmetres) of the building? (a) 50(3 +√3) (b) 100(√3 + 1 ) (c) 150 (d) 100 √3 29. The upper part of a tree broken over by the wind make an angle of 60° with the ground. The distance between the root and the point where top of the tree touches the ground is 25 metres. What was the height (in metres) of the tree? (a) 84.14 (b) 93.3 (c) 98.2 5 (d) 120.2 4

21. What is the value of [(cos 7A + cos 5A) ÷ (sin 7A – sin 5A)]? (a) tan A (b) tan 4 A (c) cot 4 A (d) cot A

30. The height of a tower is 300 meters. When its top is seen from top of another tower,then the angle of elevation is 60°. The horizontal distance betweenthe bases of the two towers is 120 metres. What is the height (in metres) of the small tower? (a) 88.24 (b) 106.7 1 (c) 92.15 (d) 112.6 4

22. What is the value of [1 – sin (90 – 2A)] / [1 + sin (90 + 2A)]? (a) sinA.cosA (b) cot²A (c) tan²A (d) sin²A.cosA

31. What is the value of [sin (y – z) + sin (y + z) + 2 sin y]/[sin (x – z) + sin (x + z) + 2 sin x]? (a) cos x sin y (b) (sin y)/(sin x) (c) sin z (d) sin x tan y 32. What is the value of {[sin (x + y) – 2 sin x + sin (x – y)]/[cos (x – y) + cos (x + y) – 2 cos x]} × [(sin 10x – sin 8x)/(cos 10x + cos 8x)]? (a) 0 (b) tan² x (c) 1 (d) 2 tan x

23. What is the value of sin 75° + sin 15°? (a) √3 (b) 2√3 3

(c)√

2

(d) 3/√2

24. What is the value of [(cos 3θ + 2cos 5θ + cos 7θ)÷(cos θ + 2cos 3θ + cos 5θ)] + sin 2θ tan 3θ? (a) cos 2θ (b) sin 2θ (c) tan 2θ (d) cot θ s in 2θ

33. What is the value of [sin (90° – 10θ) – cos (π – 6θ)]/ [cos (π /2 – 10θ) – sin (π – 6θ)]? (a) tan 2θ (b) cot 2θ (c) cot θ (d) cot 3θ

25. What is the value of [2 sin (45 + θ) sin (45 – θ)]/cos 2θ? (a) 0 (b) tan 2θ (c) cot 2θ (d) 1

34. If sec θ (cos θ + sin θ) = √2, then what is the value of (2 sin θ)/(cos θ – sin θ)? (a) 3√2 (b) 3/√2 (c) 1/√2 (d) √2

26. What is the value of sin (90° + 2A)[4 – cos² (90° – 2A)]? (a) 2(cos³A – sin³A) (b) 2(cos³A + sin³A) (c) 4(cos⁶A+ sin⁶A) (d) 4(cos⁶A – sin⁶A) 15

35. What is the value of (a) tan² θ sec² θ (c) tan⁴ θ

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1

1

sin4(90−θ)

+ [cos2(90−θ)]−1 ?

(b) sec⁴ θ (d) tan² θ sin² θ

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