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

This is a guest article by Mr.Dipak Singh.

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 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 nθ 0 +1 Can be written as0 ≤ Sin2 nθ ≤ 1 cos2 θ , cos2 3θ , cos2 8θ … cos2 nθ 0 ≤ Cos2 nθ ≤ 1 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

# 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.

For more articles on trigonometry and aptitude, visit mrunal.org/aptitude

## Mrunal recommends

Books: UPSC Prelims Books: UPSC Mains Books: Maths /Reasoning

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

1. What is the extreme values of cos 2x+cos square x

1. cos(2x) = cos^2 (x) – sin ^2 (x)
so if we convert cos(2x) and simplify , it will come as 3cos^2(x) -1
and min value of cos^2 (x) is 0,
so
min value is “-1″
max value is ” 2″
please correct me if i am wrong.

1. cos(2x)= 2cos^2(x) – 1…..NOT 3cos^2(x) – 1
so max value =1
and min value=-1

1. Cos(2x) + (Cos X)^2
= { (Cos X)^2 – (Sin X)^2 } + (Cos X)^2
= { 2 (Cos X)^2 – 1 } + (Cos X)^2
= 3 (Cos X)^2 – 1

Max value = 2
Min value = –1

1. 8sin^2x-cos^2x
=9sin^2x-1
Max.: 8
Min.:0

2. She has written direct answer after simplification. Its correct

3. 8sin^2x-cos^2x find min. and max. Value

5. Maximum and minimum value of Cos(cosx)

1. Its cos1 both min and max…

1. Max is 1 (= cos0) which is obtained when x=π/2

6. What is the maximum valu of 1-(3sin@+4cos@)……?

1. max. value of the function 1- (3 sin@ + 4 cos@)………………………….when 3sin@ + 4 cos@ is minimum
and min. value of 3sin@+ 4 cos@ = -5………………………..so max. value of 1- (3sin@+ 4cos@) =1-5 =-4

1. Bhai ye – ve function h isliye 1-(-5) =6 hona chahiye max. Neg fun m scene reverse ho jata h min value max and max value min bn jati h

7. If sec square theta = 4xy÷ (x+y)^2 holds for some pair (x,y) then how many pairs (x,y) are possible being real and x not equal to -y

8. I want to know how to find minimum and maximum values of sin^4x + cos^8x, like if there are different powers on trigonometric functions?? Please do reply:)

1. Max. Value will be 1
As sin^2m theta + cos^2n theta <=1 always .

9. hi…
what is maximum of sin(x)+a sin(2x)=???

1. Sinx has – 1 to +1
Sin2x=2sinx cosx
=2(1/2)…….{sinx.cosx has – 1/2 to +1/2}
So for maximum of value
1+2(1/2)
=2

10. How to find the minimum and maximum value of 3sinx+cosec cube x

11. 64sinA x 256cosA का उच्चतम मान ज्ञात कीजिये|

1. 8192

12. Minimum and maximum value of 1/2- cos3x

13. Find min or max value of cot^2( x ) +sec^2(x)