maximum and minimum of trignometric functions

 

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,

Min-Max table

Min value Max value Can 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 …sin2 0 +1 Can be written as0 Sin21
cos2 θ , cos2 3θ , cos2 8θ … cos2 0 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)

Short Tricks to find Max and Min Value of Trigonometric identity

Dear Readers,

We are providing you trigonometric identity shortcuts which are usually asked in IIT Mains Exams. Use these below short cuts to solve questions within minimum time.

Type-I

In case of sec2x, cosec2x, cot2x and tan2x, we cannot find the maximum value because they can have infinity as their maximum value. So in question containing these trigonometric identities, you will be asked to find the minimum values only. The typical question forms are listed below:

3

Example: -1

Find the Minimum value of 9 cos 2x + 2 sec 2x

sol – this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

  • Minimum Value = 2√ 9 x  2= 2√ 18

Example:-2

Find the Minimum value of 8 tan 2x + 7 cot 2x
sol – this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

  • Minimum Value = 2√ 8 x  7= 2√ 56

Type -II

2

Example -1

Find the Maximum and Minimum Value of 3 sin x + 4 cos y

Sol- If you find the question of this kind, apply the above formulae.

  • Maximum Value = √ 9 + 16   = √ 25   = 5
  • Minimum Value = – √ 9 + 16   = – √ 25   =  – 5

Example-2

Find the Maximum and Minimum Value of 3 sin x + 2 cos y

Sol- If you find the question of this kind, apply the above formulae.

  • Maximum Value = √ 9 + 4   = √ 13
  • Minimum Value = – √ 9 + 4   = – √ 13

   Type III

1

Example -1  

Find the maximum and Minimum Value of 3 sin 2x + 4 cos 2x

Sol- Here the 4> 3 so

  •  Maximum Value = 4
  • Minimum Value = 3

 

Example –2

Find the maximum and Minimum Value of 5 sin 2x + 3 cos 2x

Sol – Here 5>3

  • Maximum Value = 5
  • Minimum Value = 3

Type-IV

 Find the Minimum Value of  Sec 2x +  cosec 2x

Sol – 1 + tan 2x +  cosec 2x    ——————————————–(Sec 2x = 1 + tan 2x)

= 1+ tan 2x + 1 +  cot 2x ————————————————(cosec 2x = 1 + cot 2x )

=2 + tan 2x +  cot 2x—————————————————apply type-3 formula

=2 + 2 √ 1 x  1 = 2 + 2 =4

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 ? 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 + cos2θ=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 + cos2θ=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!