[Trigonometry] Finding Minimum Maximum Values for SSC CGL Made Easy without differentiation

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  1. Trig-Identities
  2. Min-Max table
  3. Ratta-fication formulas
  4. The AM GM Logic
  5. Find minimum value of 4 tan2θ + 9 cot2θ
  6. Extra facts
  7. SSC CGL 2012 Tier II Question
  8. The least value of 2 sin2 θ + 3 cos2 θ (CGL2012T1)
  9. The maximum value of Sin x + cos x  is 
  10. The maximum value of 3 Sin x – 4 Cos x  is 
  11. Min Max values of  sin 4x + 5 are
  12. Minimum and maximum value of  Sin Sin x  is

Today we’ll see how to find the maximum value (greatest value ) or the minimum value (least value) of a trigonometric function without using differentiation. Take a pen and note-book, keep doing the steps while reading this article.
First Remember following identities:

Trig-Identities

1. sin2 θ + cos2 θ = 1

2. 1+ cot2 θ = cosec2 θ

3. 1+ tan2 θ = sec2 θ

how did we get these formulas? Already explained, click me

Min-Max table

Min valueMax valueCan be written as
sin θ, sin 2θ, sin 9θ …. sin nθ-1+1-1 Sin nθ 1
cos θ, cos 4θ , cos 7θ … cos nθ-1 Cos nθ 1
sin2 θ , sin2 4θ , sin2 9θ …sin20+1Can be written as0 Sin21
cos2 θ , cos2 3θ , cos2 8θ … cos20 Cos21
Sin θ Cos θ-1/2+1/2-1/2 ≤ Sin θ Cos θ ≤ ½

observe that in case of sin2θ and cos2θ, the minimum value if 0 and not (-1). Why does this happen? because (-1)2=+1

Negative Signs inside out

  • Sin (- θ) = – Sin (θ)
  • Cos (-θ) = Cos (θ)

Ratta-fication formulas

  1. a sin θ ± b cos θ =  ±√ (a2 + b2 ) { for min. use – , for max. use + }
  2. a sin θ ± b sin θ =   ±√ (a2 + b2 ) { for min. use – , for max. use + }
  3. a cos θ ± b cos θ =  ±√ (a2 + b2 ) { for min. use – , for max. use + }
  4. Min. value of (sin θ cos θ)n = (½)n

The AM GM Logic

Let A ,B are any two numbers then,

Arithmetic Mean (AM)= (A + B) / 2 and

Geometric Mean (GM) = √ (A.B)

  • Hence, A.M ≥ G.M  ( We can check it by putting any values of A and B )
  • Consider the following statement “ My age is greater than or equal to 25 years . ”
  • What could you conclude about my age from this statement ?
  • Answer : My age can be anywhere between 25 to infinity … means it can be 25 ,  , 50 ,99,  786 or 1000 years etc… but it can not be 24 or 19 or Sweet 16 . Infact it can not be less than 25, strictly.
  • Means, We can confidently say that my age is not less 25 years. Or in other words my minimum age is 25 years.

Showing numerically, if Age ≥ 25 years ( minimum age = 25 )

  • Similarly, If I say x ≥ 56 ( minimum value of x = 56 )
  • If, y ≥ 77 ( minimum value of y = 77 )
  • If, x + y ≥ 133 ( minimum value of x + y = 133 )
  • If, sin θ  ≥ – 1 ( minimum value of Sin θ = -1 )
  • If, tan θ + cot θ ≥ 2 (minimum value of tan θ + cot θ = 2 ) ]]

Sometimes, we come across a special case of trigonometric identities like to find min. value of sin θ + cosec θ or tan θ + cot θ or  cos2 θ + sec2 θ etc. These identities have one thing in common i.e., the first trigonometric term is opposite of the second term or vice-versa ( tan θ = 1/ cot θ , sin θ = 1/ cosec θ , cos2 θ = 1/ sec2 θ ).

These type of problems can be easily tackled by using the concept of

A.M ≥ G .M

Meaning, Arithmetic mean is always greater than or equal to geometric mean. For example:

Find minimum value of 4 tan2θ + 9 cot2θ

(they’ll not ask maximum value as it is not defined. )

We know that tan2θ = 1/ cot2θ , hence applying A.M ≥ G.M logic, we get

A.M of given equation = (4 tan2θ + 9 cot2θ) / 2 …. (1)

G.M of given equation =  √ (4  tan2θ . 9 cot2θ )

=  √ 4 * 9   # ( tan2θ and cot2θ inverse of each other, so tan x cot =1)

= √ 36 = 6 …. (2)

Now, we know that A.M ≥ G. M

From equations (1) and (2) above we get,

=> (4 tan2 θ + 9 cot2θ) / 2  ≥ 6

Multiplying both sides by 2

=>  4 tan2 θ + 9 cot2 θ ≥ 12 ( minimum value of tan2 θ + cot2 θ is 12 )

Deriving a common conclusion:

  • Consider equation  a cos2 θ + b sec2 θ ( find minimum value)
  • As, A.M ≥ G.M
  • (a cos2 θ + b sec2 θ / 2 ) ≥  √ (a cos2 θ . b sec2 θ)
  • a cos2 θ + b sec2 θ ≥ 2 √ (ab) ( minimum value 2 √ab )
  • So, we can use 2 √ab directly in these kind of problems.

Summary:

While using A.M ≥ G.M logic :

  • Term should be like a T1 + b T2 ; where  T1 = 1 / T2
  • Positive sign in between terms is mandatory. (otherwise how would you calculate mean ? )
  • Directly apply  2√ab .
  • Rearrange/Break terms if necessary -> priority should be given to direct use of identities ->  find terms eligible for A.M ≥ G.M logic -> if any, apply -> convert remaining identities, if any, to sine and cosines -> finally put known max., min. values.

Extra facts:

  • The reciprocal of 0 is + ∞ and vice-versa.
  • The reciprocal of 1 is 1 and -1 is -1.
  • If a function has a maximum value its opposite has a minimum value.
  • A function and its reciprocal have same sign.

Keeping these tools (not exhaustive) in mind we can easily find Maximum or Minimum values easily.

SSC CGL 2012 Tier II Question

What is The minimum value of sin2 θ + cos2 θ + sec2 θ + cosec2 θ + tan2 θ + cot2 θ

  1. 1
  2. 3
  3. 5
  4. 7

Solution:

We know that sin2 θ + cos2 θ = 1 (identitiy#1)

Therefore,

(sin2 θ + cos2 θ) + sec2 θ + cosec2 θ + tan2 θ + cot2 θ

= (1) + sec2 θ + cosec2 θ + tan2 θ + cot2 θ

Using A.M ≥ G.M logic for tan2 θ + cot2 θ we get ,

= 1 + 2 +  sec2 θ + cosec2 θ

changing into sin and cos values

( Because we know  maximum and minimum values of Sin θ, Cos θ :P and by using simple identities we can convert all trigonometric functions into equation with Sine and Cosine.)

=  1 + 2 + (1/ cos2 θ) + (1/ sin2 θ)

solving taking L.C.M

= 1 + 2 +  (sin2 θ + cos2 θ)/( sin2 θ . cos2 θ)…..eq1

but we already know two things

sin2 θ + cos2 θ=1 (trig identity #1)

Min. value of (sin θ cos θ)n = (½)n (Ratta-fication formula #4)

Apply them into eq1, and we get

= 1 + 2 + (sin2 θ + cos2 θ)/( sin2 θ . cos2 θ)

= 1 + 2 + (1/1/4) = 1+2+4

= 7 (correct answer D)

The least value of 2 sin2 θ + 3 cos2 θ (CGL2012T1)

  1. 1
  2. 2
  3. 3
  4. 5

We can solve this question via two approaches

Approach #1

Break the equation and use identity no. 1

= 2 sin2 θ + 2 cos2 θ + cos2 θ

=2(sin2 θ + cos2 θ) + cos2 θ ; (but sin2 θ + cos2 θ=1)

= 2 + cos2 θ ;(but as per min-max table, the minimum value of cos2 θ=0)

= 2 + 0 = 2 (correct answer B)

Approach #2

convert equation into one identity ,either sin or cos

first convert it into a sin equation :

= 2 sin2 θ + 3 (1- sin2 θ) ;(because sin2 θ + cos2 θ=1=>cos2 θ=1- sin2 θ)

= 2 sin2 θ + 3 – 3 sin2 θ

= 3 – sin2 θ

= 3 – ( 1) = 2 (but Min. value of sin2 θ  is 0 …confusing ???? )

As sin2 θ is preceded by a negative sign therefore we have to take max. value of  sin2 θ in order to get minimum value .

Converting into a cos equation :

= 2 sin2 θ + 3 cos2 θ

= 2 (1- cos2 θ) + 3 cos2 θ

= 2 – 2 cos2 θ + 3 cos2 θ

= 2 + cos2 θ

= 2 + 0 = 2 ( correct answer B )

The maximum value of Sin x + cos x  is

  1. √2
  2. 1/ √2
  3. 1
  4. 2

Applying Ratta-fication formulae No.1

a sin θ ± b cos θ =  ±√ (a2 + b2 ) { for min. use – , for max. use + }

in the given question, we’ve to find the max value of

Sin x + cos x
= + √ (12+ 12 )

= √2 ( correct answer A )

The maximum value of 3 Sin x – 4 Cos x  is

  1. -1
  2. 5
  3. 7
  4. 9

Solution:

Applying Ratta-fication formulae No.1

a sin θ ± b cos θ =  ±√ (a2 + b2 ) { for min. use – , for max. use + }

in the given question, we’ve to find the max value of

3 Sin x – 4 Cos x
= + √ (32+ 42 )

= √25

= 5  ( correct answer B )

Min Max values of  sin 4x + 5 are

  1. 2, 6
  2. 4, 5
  3. -4, -5
  4. 4, 6

Solution:

We know that, -1 Sin nx 1

= -1 ≤ Sin 4x ≤ 1

Adding 5 throughout, 4 ≤ Sin 4x +5 ≤ 6

Therefore, the minimum value is 4 and maximum value is 6 ( correct answer D )

Minimum and maximum value of  Sin Sin x  is

  1. Do not exist
  2. -1, 1
  3. Sin -1  , Sin +1
  4. – Sin 1 , Sin 1

We know that, -1 Sin nx 1

= Sin (-1) ≤ Sin x Sin (1)

=  – Sin 1 ≤ Sin x  ≤ Sin 1 ;  [Sin(-θ) is same as – Sin θ ]

Therefore, Minimum value is –Sin 1 and maximum is Sin 1 ( correct answer D)

The key to success is Practice! Practice! Practice!

Drop your problems in the comment box.

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287 Comments on “[Trigonometry] Finding Minimum Maximum Values for SSC CGL Made Easy without differentiation”

  1. If ‘m’ and ‘M’ are the minimum and maximum values of 4+1/2 (sinx)^2 -2 (cosx)^4, x∈R, then M-m is equal to

    1. my answer is M-m=1
      m=2 and M=3

  2. given the expression sin2x.cos2x

    1.calculate the maximum value of the above expression
    2.calculate the first negative value of x for which the expression has a maximum value

    1. 1. Sin2x.cos2x=1÷2(2sin2x.cos2x)
      =0.5(sin4x)
      Max = 0.5 × {max of sin4x=1)
      =Max value is 0.5
      2. Solve this by sin 4x graph
      Period will be pie/2
      Ans=3pie/8

  3. Find the max and min value of sin^2A + cos^4A

    1. Max value-1 and Min. value-3/4 (By putting A= 45 degrees)…. :)

    2. Given
      Sin^2x+cos^4x
      1-cos^2x+cos^4x
      1-cos^2x(1-cos^2x)
      1-cos^2xsin^2x
      Minima of cos^2xsin^2x =1/4
      So minima of function = 3/4
      Maxima=5/4

  4. How to solve 1+2sinx + 3cos^2

  5. How to solve 1+2sinx + 3cos^2 ?
    Find its min and max values

  6. I have a question. What is the minimum value of 2^sinx+2^cosx?

    1. amit kumar das (the minimum value willbe 2^1+1/2^1/2

  7. What is the answer is M-m=1
    m=2 and M=1

  8. IF THE STANDARD DEVIATION OF NUMBERS 2,3,A AND 11 IS 3.5,THEN WHICH OF THE FOLLOWING IS TRUE
    a)3A^2-23A+44=0
    B)3A^2-26A+55=0
    C)3A^2-34A+91=0
    D)3A^2-321A+84=0

  9. a sin θ ± b sin θ = ±√ (a2 + b2 ) { for min. use – , for max. use +
    Sir in the following equation why can’t we directly solve taking sin theta as common so we have direct result a-b or a + b after solving this to get the maximum value we can take sin theta is equal to +1 to get the minimum value sin theta is equal to -1

  10. Can any1 tell me how M=3?

  11. In the Ratta-Fiction formula it has been given as —- Min. value of (sin θ cos θ)n = (½)n

    Min. value of (sin θ cos θ)n = (½)n–If the value of n is 1, the minimum value of sinθCosθ will be 1 which is contrary to the
    value given in the table (-1/2 ≤ Sin θ Cos θ ≤ ½).
    In Min-Max table Sin θCos θ minimum value is given as -1/2 (-1/2 ≤ Sin θ Cos θ ≤ ½).

    Requesting you to clarify.

  12. who formulate this ratta law of fiction

    1. omg! its not the ratta law of fiction ! its called atta-fication. have you ever heard of ‘ratta-maar’ in hindi? so thats converted into an abstract noun: ‘ratta-fication’, just for a joke! its basically called “formulas which you gotta mug up”

  13. Find the range of 3sinx+4cosx

  14. find the maximum and minimum value of (2+3^1/2)sin(theata) + 3^1/2cos(theata)????

  15. Prove the rattafication formulae using graphs

  16. What is the maximum and minimum value of sin(cos x)

  17. What is the minimum value of (CosA)^4+(SinA)^2

  18. What is the tan^2x+ cot^2x?

  19. A function is given as
    y=6sinA+8cosA

    a) Express y=6sinA+8cosA in the form of r sin (A+B) where B is an acute angle.
    b) Give the cordinates of the MAXIMUM and MINIMUM values of the function y=6sinA+8cosA

  20. What is maxima and minima of sinx+sin2x+sin4x+sin5x

  21. I want the proof for finding the maximun and minimum values if acosx+bsinx+c

  22. Maxima =√a^2+b^2 + C
    Minima= -√a^2+b^2 + C

  23. Find the maximum value of {cos^2(x) + cos^2(y) – cos^2(z)}.

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