Trigonometry MCQ - It contains important 25 questions that are fully solved. Each question is of PDF

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It contains important 25 questions that are fully solved. Each question is of MCQ type so that it can help you in any entrance exams...

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Subject:-Mathematics Topic:-Trigonometric MCQ Level:-All Entrances Exams

1. What is the measure of the angle 1140 35′ 30′′ in radian?

a) 1 rad

b) 2 rad

c) 3rad

d) 4rad

A. The answer is b) 2rad

′

71 ′

soln:- 35′ 30′′ = (35 + 2) = ( 2 ) 1

71 ′

⇨(2) =(

71

2

1 0

0

71

× 60) = ( 120)

71 0 13751 0 ∴ 114 35 30 = (114 + ) =( ) 120 120 0

′

′′

We know that , 2𝜋 𝑟𝑎𝑑 = 3600 13751 0

⇨(

120

) =

2𝜋

3600

×

13751

= 2.00008069 𝑟𝑎𝑑 = 2 𝑟𝑎𝑑 (𝑎𝑝𝑝𝑟𝑜𝑥).

120

10

2. What is the value of sin 292 2 ? 1 a) √2 + √3 3

c) 2 √2 + √2 1

A. The answer is d) − 2 √2 + √2 1

10

585 0

soln:- sin (292 2 ) = sin (

2

)

b) −

1

d) −

3

1

2

√2 − √3

√2 + √2

= −√

(1−𝑐𝑜𝑠5850 )

= −√

√2+1

3. tan

7𝜋

2√2

6

2

= −√ 9𝜋

, tan

4

= −√

√2+1 2√2

1−𝑐𝑜𝑠2250

×

10𝜋 3

, tan

√2

√2

2

= −√

1+𝑐𝑜𝑠450 2

= − √(√2 + 2) 1

= −√

1√2 (1+2

)

2

are in?

a) AP

b) GP

c) HP

d) None of these

A. The answer is b) GP soln:- tan tan

tan

9𝜋 4

7𝜋 6

1

= tan (2𝜋 + 4 ) = tan 4 = 1

10𝜋 3

Clearly

= tan (2𝜋 + 1

√3

𝜋

𝜋

4𝜋

) = tan 3

4𝜋 3

, 1, √3 are in GP.

4. If 𝑐𝑜𝑠𝜃 = a) √

𝜋

𝜋

= tan (𝜋 + 6 ) = tan 6 = √3

𝑎𝑐𝑜𝑠𝜑+𝑏

𝜑

c) √ 𝑎+𝑏 sin

𝜑

𝜃

b) √

tan 2 𝑎+𝑏 2

𝑎+𝑏

c) None of these 𝑎−𝑏

𝜑

𝑎𝑐𝑜𝑠𝜑+𝑏

𝑎+𝑏𝑐𝑜𝑠𝜑

Applying componendo and dividendo rule, 1−𝑐𝑜𝑠𝜃 1+𝑐𝑜𝑠𝜃

=

𝜑

cos 2 𝑎−𝑏

A. The answer is a) √ 𝑎+𝑏 tan 2 soln:- Now, 𝑐𝑜𝑠𝜃 =

𝜋

𝜋

, then tan 2 is equal to? 𝑎+𝑏𝑐𝑜𝑠𝜑

𝑎−𝑏

𝑎−𝑏

= tan (𝜋 + 3 ) = tan ( 3 ) = √3

[𝑎(1−𝑐𝑜𝑠𝜑)−𝑏 (1−𝑐𝑜𝑠𝜑)] 𝑎(1+𝑐𝑜𝑠𝜑)+𝑏 (1+𝑐𝑜𝑠𝜑)

⇨ 𝑡𝑎𝑛2 = 𝑎−𝑏 𝑡𝑎𝑛2 𝜑2 2 𝑎+𝑏 𝜃

𝜃

⇨ tan = √𝑎−𝑏 tan 𝜑2 𝑎+𝑏 2

5. If 𝐴 + 𝐵 = 90°, then minimum and maximum values of (𝑐𝑜𝑠𝐴 𝑐𝑜𝑠𝐵) respecrively are 1 1

a) − , 4 4

b) −

1 1

c) − 2 , 2

1

, 3

1

3

d) None of these 1 1

A. The answer is c) − 2 , 2

soln:- 𝑐𝑜𝑠𝐴 𝑐𝑜𝑠𝐵 = 2 (𝑐𝑜𝑠𝐴 𝑐𝑜𝑠𝐵) 1

= 2 [cos(𝐴 + 𝐵) + cos(𝐴 − 𝐵)] 1

= 2 [𝑐𝑜𝑠90° + cos(𝐴 − 𝐵 )] 1

= 2 cos (𝐴 − 𝐵) 1

∵ −1 ≤ cos (𝐴 − 𝐵) ≤ 1 1

⇨ − ≤ cos (𝐴 − 𝐵) ≤ 2 2 1

6. If 𝑥, 𝑦 and 𝑧 are the angles of a triangle and 𝑧 = 135°. Then what is the value of (1 + tan 𝑥)(1 + tan 𝑦)? a) 1

b) 2

c) 3

A. The answer is b) 2

soln:- We have 𝑥 + 𝑦 + 𝑧 = 180°

⇨ 𝑥 + 𝑦 = 180° − 𝑧 = 180° − 135° = 45°

∴

⇨

tan(𝑥 + 𝑦) = tan(45°) = 1

𝑡𝑎𝑛𝑥+𝑡𝑎𝑛𝑦

1−𝑡𝑎𝑛𝑥 𝑡𝑎𝑛𝑦

=1

d) 4

⇨ tan 𝑥 + tan 𝑦 = 1 − tan 𝑥 tan 𝑦

⇨ tan 𝑥 + tan 𝑥 tan 𝑦 + tan 𝑦 = 1 ………….(i)

Now, (1 + tan 𝑥)(1 + tan 𝑦) = 1 + tan 𝑥 + tan 𝑦 + tan 𝑥 tan 𝑦

=1 + 1 = 2……………[by (i)]

7. Let 𝑡𝑎𝑛²𝑥 = 1 − 𝑒², 𝑒 is any constant, then the value of (𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛3 𝑥 𝑐𝑜𝑠𝑒𝑐𝑥) is

a) (2 + 𝑒 2 )3/2

b) (2 − 𝑒 2 )3/2

c) (1 − 𝑒 2 )3/2

A. The answer is b) (2 − 𝑒 2 )3/2

d) (1 + 𝑒 2 )3/2

soln:- 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛3 𝑥 𝑐𝑜𝑠𝑒𝑐𝑥 = 𝑠𝑒𝑐𝑥 (1 + 𝑡𝑎𝑛3 𝑥

𝑐𝑜𝑠𝑒𝑐𝑥

= sec 𝑥 (1 + 𝑡𝑎𝑛2 𝑥)

𝑠𝑒𝑐𝑥

)

=sec 𝑥 (𝑠𝑒𝑐²𝑥)

= 𝑠𝑒𝑐³ 𝑥 = (𝑠𝑒𝑐 2 𝑥)3/2 = (1 + 𝑡𝑎𝑛2 𝑥)3/2 = (2 − 𝑒²)3/2

8. Which one of the following is positive in the third quadrant? a) sinθ

b) cosθ

A. The answer is c) tanθ

c) tanθ

d) secθ

5

9. What is the value of 𝑠𝑒𝑐² {𝑡𝑎𝑛−1 ( )}? 11 a) 121/96

b) 217/921

A. The answer is c) 146/121 soln:- Given: 𝑠𝑒𝑐² {𝑡𝑎𝑛−1 ( 11)} 5

c) 146/121

d) 267/121

5

= 1 + 𝑡𝑎𝑛² {𝑡𝑎𝑛−1 ( )} = 1 + [tan {𝑡𝑎𝑛−1 ( 5 )}] ² 11 11 5

= 1 + ( ) ² = 1 + 25 = 146 121 121 11

10. What is the value of 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐴 𝑡𝑎𝑛𝐴 + 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 𝑐𝑜𝑡𝐴?

a) 𝑠𝑖𝑛𝐴

b) 𝑐𝑜𝑠𝐴

c) 𝑡𝑎𝑛𝐴

A. The answer is d) 1

d) 1

soln:- We have, 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐴 𝑡𝑎𝑛𝐴 + 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 𝑐𝑜𝑡𝐴 = 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐴

𝑠𝑖𝑛𝐴

𝑐𝑜𝑠𝐴

𝑐𝑜𝑠𝐴

+ 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐴

= 𝑠𝑖𝑛2 𝐴 + 𝑐𝑜𝑠 2 𝐴 = 1

11. If θ=18°, then what is the value of 4𝑠𝑖𝑛²𝜃 + 2𝑠𝑖𝑛𝜃? a) −1

b) 1

c) 0

d) 2

A. The answer is b) 1 soln:- Given, θ=18°

Now, we have 4𝑠𝑖𝑛²𝜃 + 2𝑠𝑖𝑛𝜃 = 4𝑠𝑖𝑛2 (18°) + 2sin (18°) = 4{

=

√5−1 }² 4

3−√5 2

+

+2{

√5−1 2

√5−1

=1

4

}=

4(5+1−2√5) 16

+

√5−1 2

12. Consider the following statements I. 1° in radian measure is less than 0.02radians. II. 1 radian in degree measure is greater than 45°. Which of the above statement(s) is /are correct? a) Only I

b) Only II

c) Both I and II

d) Neither I nor II

A. The answer is b) Only II 𝜋

radian = which is equal to 0.02

soln:- I. 1°=

180

II. 1 radian =

180 𝜋

= 0.017 = 0.02(approx)

3.14 180

degree =

which is greater than 45°

180 3.14

= 57.32 degree

13. Let sin(𝐴 + 𝐵 ) = 1 and sin(𝐴 − 𝐵) = 2, where A,B ∈ [0, 2 ], then what is the value of 𝑠𝑖𝑛²𝐴 − 𝑠𝑖𝑛²𝐵? a) 0

𝜋

1

b) ½

c) 1

d) 2

A. The answer is b) ½

soln:- Given: sin(𝐴 + 𝐵) = 1, ⇨ sin(𝐴 + 𝐵) = sin 𝜋

𝜋

2

⇨ 𝐴 + 𝐵 = …………………(i) 2 and sin(𝐴 − 𝐵) =

1

2

⇨ sin(𝐴 − 𝐵) = sin 3

𝜋

⇨ 𝐴 − 𝐵 = 3 ……………(ii) 𝜋

Solving (i) and (ii), we get A=π/3 and B=π/6 𝜋

Therefore 𝑠𝑖𝑛²𝐴 − 𝑠𝑖𝑛²𝐵 = 𝑠𝑖𝑛2 ( ) − 𝑠𝑖𝑛2 ( ) 3 6 𝜋

= ( )²− ( )² = − = 2 4 4 2 2 √3

14. If 𝑠𝑒𝑐 ∝= a) 5/13

3

1

13 5

1

1

, where 270°...