Trigonometry

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Trigonometry


  • Trigonometry: In a right angled OAB, where BOA = ,
i.   sin  = Perpendicular = AB ;
Hypotenuse OB

 

ii.   cos  = Base = OA ;
Hypotenuse OB

 

iii.  tan  = Perpendicular = AB ;
Base OA

 

iv.  cosec  = 1 = OB ;
sin AB

 

v.   sec  = 1 = OB ;
cos OA

 

vi.  cot  = 1 = OA ;
tan AB
  • Trigonometrical Identities:
    1. sin2  + cos2  = 1.
    2. 1 + tan2 = sec2 .
  • 1 + cot2 = cosec2 .

 

Values of T-ratios:

(/6)

30°

(/4)

45°

(/3)

60°

(/2)

90°

sin 0
1
2
3
2
1
cos 1
3
2
1
2
0
tan 0
1
3
1 3 not defined
  • Angle of Elevation:

Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, the angle which the line of sight makes with the horizontal through O, is called the angle of elevation of P as seen from O.

Angle of elevation of P from O = AOP.

 

  • Angle of Depression:

Suppose a man from a point O looks down at an object P, placed below the level of his eye, then the angle which the line of sight makes with the horizontal through O, is called the angle of depression of P as seen from O.

 

Ques1. The maximum and minimum value of 8 sin  and cos  + 4 cos 2 is:

  • 4 and -4 ) 16 and -16                         c.) 4√2 and -4√2                         d.) 8 and -8                        e.) None of these

Explanation: Answer is C

Here, 8 sin cos + 4 cos 2 = 4 sin 2 + 4 cos 2

So, maximum value =  = 4√2 and minimum value = – = -4√2

Ques2. Solve cos + sin = √2, then the value of  is:

  • )                             c.)                            d.)                                e.) None of these

Explanation: Answer is B

cos + sin = √2

+  = 1      (∵ divded throughout by √2)

= 1

cos() = 1 = cos0

∴  = 0

  • =

Ques3. The value of  is

  • 2 + 3√2        ) 2 – 3√2                         c.) 3 – 2√2                         d.) 3 + 2√2                  e.) None of these

Explanation: Answer is D

sin  = sin() = sin =

cos 120 = cos(180 = -cos60 = –

∴  =  =  *  =  = 2+1+2 = 3 + 2√2

Ques4. The value of  is

  • 1 ) ∞                      c.) 0                         d.) -1                        e.) None of these

Explanation: Answer is B

As,  = tan(A + B)

= tan(47° + 43°) = tan90° = ∞

Ques5. If A – B = , then the value of cos A cos B + sin A sin B is

  • ½         ) 1                         c.) 3/2                           d.) 0                         e.) None of these

Explanation: Answer is A

As, cos A cos B + sin A sin B = cos(A – B) = cos =                 (∵ A – B = )

Ques6. If A + B = , then the value of (cot A – 1) (cot B – 1) is

  • 1                         ) -1                          c.) 0                               d.) 2                            e.) None of these

Explanation: Answer is D

As, (cot A – 1) (cot B – 1) =   =      (∵ A + B = 45

=   =

=  = 2

Ques7. The value of 3 sin 15 – 4 sin3 15 is:

  • 1 ) ½                           c.) 1/√2                        d.) 0                          e.) None of these

Explanation: Answer is C

As, 3 sin – 4 sin3 = sin 3

So, 3 sin 15 – 4 sin3 15 = sin 3(15) = sin  =

Ques8. The maximum value of sin x + cos x is:

  • -√2 ) √2                         c.) 1                           d.) -1                            e.) None of these

Explanation: Answer is B

The maximum value of a sin + b cos is

So, here a = 1, b = 1

Maximum value =  = √2

 

 

Ques9. If sin  = , then cos is:

  • )                           c.)                           d.)                           e.) None of these

Explanation: Answer is C

sin  =  =

C

 

3                      5

 

 

B                                   A

By Pythagoras Theorem,

H2 = P2 + B2

52 = 32 + B2

B2 = 16

B = 4

cos =  =

Ques10. If tan C = 11, then sin2 C + cos2 C is equal to

  • 2 )                           c.) 0                             d.) 1                          e.) None of these

Explanation: Answer is D

If tan C = 11, sin2 C + cos2 C = 1 it is an identity.


Ques11. If sin  = , then the value of tan  is equal to

  • )                        c.)                          d.)                     e.) None of these

Explanation: Answer is B

sin  = , here AC =

C

 

 

2ab

 

B                      A

 

BC = 2ab

∴ AB2 = AC2 – BC2 = (a2 + b2)2 – (2ab)2 = a4 + b4 + 2a2b2 – 4 a2b2

= a4 + b4 + 2a2b2 = (a2 – b2)2

    ∴ AB = a2 – b2           ∴ tan  =

Ques12. How many degrees are there in an angle which equals two-third of its complement?

  • 36 ) 45                          c.) 48                        d.) 60                       e.) None of these

Explanation: Answer is A

Given, α + β = 90          …(i)

By given condition,

β = α

∴ β = (90)                      [from eq. i]

  • β = 60β => β = 36

Ques13. If sin B = , then 3 cos B – 4 cos3 B is equal to

  • 1 )                          c.) 0                            d.)                           e.) None of these

Explanation: Answer is C

sin B = , then cos B = B = =

∴ 3 cos B – 4 cos3 B = 3 – 4

=  –  =  –  = 0

Ques14. If sec α = , then  is equal to

  •                        ) sin α cos α                        c.) cos 2α                     d.) tan α                         e.) None of these

Explanation: Answer is B

=

=  *  = sinα . cosα

Ques15. What is the value of cos 1 cos 2 cos 3….cos 90

  • ) 0                          c.) 1                            d.) 2                         e.) None of these

Explanation: Answer is B

∵ cos 90 = 0

∴ cos 1 cos 2 cos 3….cos 90 = 0

Ques16. sin(A + B) + sin(A – B) is equal to

  • 2 sin A cos B ) 2 cos A sin B                         c.) 2 cos A cos B                           d.) 2 sin A sin B                       e.) None of these

Explanation: Answer is A

sin(A + B) + sin(A – B) = (sin A cos B + cos A sin B) + (sin A cos B – cos A sin B)

= 2 sin A cos B

Ques17. The value of cos 15  – sin 15 is equal to

  • )                          c.)                           d.)                            e.) None of these

Explanation: Answer is B

cos 15  – sin 15 = cos 15  – sin(90 – 75)

= cos 15 – cos 75 = 2 sin * sin  = 2 sin * sin  = 2 *  =

Ques18. If 8 sin x = 4 + cos x, the values of sin x are

  • , )  ,                       c.) ,                          d.) ,                        e.) None of these

Explanation: Answer is C

8 sin x – 4 = cos x =

(8 sin x – 4)2 = 1 – sin2 x

  • 64 sin2 x + 16 – 64 sin x + sin2 x – 1 = 0
  • 65 sin2 x – 64 sin x + 15 = 0
  • (5 sin x – 3) (13 sin x – 5) = 0
  • sin x = , sin x =

Ques19. What is the value of sin2 15 + sin2 20 + sin2 25 + …+ sin2 75?

  • tan2 15+ tan2 20 + tan2 25 + …+ tan2 75
  • cos2 15+ cos2 20 + cos2 25 + …+ cos2 75
  • cot2 15+ cot2 20 + cot2 25 + …+ cot2 75
  • sec2 15+ sec2 20 + sec2 25 + …+ sec2 75

Explanation: Answer is B

sin2 15 + sin2 20 + sin2 25 + …+ sin2 75 = sin2 (90) + sin2 (90) +…+  sin2 (90)

= cos2 15 + cos2 20 + cos2 25 + …+ cos2 75

Ques20. If tan A = 1 and tan B = , then cos A.cos B – sin A.sin B is equal to

  • )                           c.)                           d.) 1                        e.) None of these

Explanation: Answer is B

tan A = 1, sin A =  =

cos A =  =

tan B = , sin B =  =

cos B =  =

∴ cos A.cos B – sin A.sin B =  *  –  *  =

Ques21. If 3 tan = 4, then  is equal to

  • )                            c.)                          d.) 1                         e.) None of these

Explanation: Answer is C

3 tan = 4 => tan =

=  –  =  –

–  =  –  =  –  =

Ques22. Given that 16 cot  = 12, then  is equal to

  • 7 ) -7                       c.)                           d.)                           e.) None of these

Explanation: Answer is A

16 cot  = 12 => cot  =  =

=  =  =  = 7

Ques23. What is the value of cot 15 cot 20 cot 70 cot 75?

  • -1 ) 0                          c.) 1                            d.) 2                          e.) None of these

Explanation: Answer is C

cot 15 cot 20 cot 70 cot 75 = tan (90) tan (90) cot 70

= tan 75 tan 70  .  = 1

= 22

tan2 +  + 2 = 4

tan2 +  = 2

Ques24. If  = 2, then the value of tan2 +  is equal to

  • 6 ) 4                           c.) 2                            d.) 3                        e.) None of these

Explanation: Answer is C

∵  = 22

tan2 +  + 2 = 4

tan2 +  = 2

Ques25. The value of sin 18 is

  • )                        c.)                           d.)                       e.) None of these

Explanation: Answer is C

Let

∴ 5 => (2+ 3) =

  • 2=  – 3
  • sin 2= sin(903)
  • 2 sincos  = cos 3
  • 2 sincos  = 4 cos3 – 3 cos
  • 2 sin = 4 cos2 – 3
  • 2 sin = 4(1 – sin2) – 3
  • 2 sin = 1 – 4 sin2
  • 4 sin2+ 2 sin  – 1 = 0
  • sin =  =

But as   = 18 lies in Ist quadrant, so it should be positive.

  • sin18 =  =