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

This is a guest article by Mr.Dipak Singh.if the math-equations/signs are not clearly visible in your browser, then click me to download the PDF version of this article.

Today we’ll see how to find the maximum value (greatest value ) or the minimum value (least value) of a trigonometric functionwithout using differentiation. Take a pen and note-book, keep doing the steps while reading this article.First Remember following 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 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 nθ0 +1

Can be written as

0 ≤ 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

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

a sin θ ± b cos θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }1.

a sin θ ± b sin θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }2.

a cos θ ± b cos θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }3.

Min. value of (sin θ cos θ)n = (½)n4.

Let A ,B are any two numbers then,

HOME APTITUDE JULY 19TH, 2013 19 COMMENTS

[Trigonometry] Finding Minimum Maximum Values for SSC CGL Made... http://mrunal.org/2013/07/trigonometry-finding-minimum-maximum-val...

1 of 7 20-Jul-13 9:00 AM

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 cannot 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 orvice-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:

(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 )

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.

While using A.M ≥ G.M logic :

Term should be like a T1 + b T2 ; where T1 = 1 / T2Positive 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.


2 of 7 20-Jul-13 9:00 AM

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.

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

1A.3B.5C.7D.

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 alltrigonometric 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)

1A.2B.3C.5D.

We can solve this question via two approaches

Break the equation and use identity no. 1

= 2 sin2 θ + 2 cos2 θ + cos2 θ


3 of 7 20-Jul-13 9:00 AM

=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)

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 )

√2A.1/ √2B.1C.2D.

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 )

-1A.5B.7C.9D.

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


4 of 7 20-Jul-13 9:00 AM

= + √ (32+ 42 )

= √25

= 5 ( correct answer B )

2, 6A.4, 5B.-4, -5C.4, 6D.

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 )

Do not existA.-1, 1B.Sin -1 , Sin +1C.- Sin 1 , Sin 1D.

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

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19 comments to [Trigonometry] Finding Minimum Maximum Values for SSC CGL Made Easy withoutdifferentiation

yo yoReply to this comment

nice explain thanks sir ji


5 of 7 20-Jul-13 9:00 AM

RajkumarReply to this comment

It is a great tutorial very nicely explained. Thanks a Lot to bothMr. Mrunal and Mr. Deepak

With Regards

rakaReply to this comment

thanks Dipak jee

Bipin DasReply to this comment

Ratta-Fication formulae

2nd nd 3rd

extreme values will be simply +(a+b) and -(a-b)

amarjitReply to this comment

thanks a lot…its very helpful

PratulReply to this comment

thanks alot sir

AyushReply to this comment

Ratta-fication formulasMin. value of (sin θ cos θ)n = (±½)n (he forgot to put ±)for example min value of sin(x)*cos(x)=[2sin(x)*cos(x)]/2=[sin(2x)]/2min value of sin2(x)=-1so min value is -1/2

correction in 2,3 has already been pointed out by Bipin Das .

Bhavna TanwarReply to this comment

Thanks Sir :)

bscvpavanReply to this comment

thanks sir….it helps me alot…

bscvpavanReply to this comment

sir ….how to find min & max value of tan x + sec x ?ans : 0 & 2.

are these correct answers?

mathematicianReply to this comment

your answers are wrong ,this trigonometric expression will not have either maximum or minimum value,the range will bebetween minus infinity to +infinity,anyway you can also think in a simple manner range of secx is,>=1 or <= -1 ,so in any case itcan not be 0.

HK


6 of 7 20-Jul-13 9:00 AM

Reply to this comment

answer = max = infinity

min = – infinity

mathematicianReply to this comment

improve your knowledge of mathematics,infinity is not a number or value ,u can’t plot this number or graph paper so bettersay it will not has either maximum value or minimum value……

TEZAReply to this comment

When SSC CGLE 2013 prelims result is going to be declared???

TEZAReply to this comment

Any clue, when SSC CGLE 2013 prelims result will be declared???

vishal kumarReply to this comment

thanx sir:)

RavikantReply to this comment

= 1 + 2 + (sin^2 θ + cos^2 θ)/( sin^2 θ . cos^2 θ)To find min value of it we have to put mini value of = (sin^2 θ + cos^2 θ)/( sin^2 θ . cos^2 θ)Therefore we have to put1st min value of (sin^2 θ + cos^2 θ) =I.e. 12nd Max value of ( sin^2 θ . cos^2 θ)=I.e. ?

maheshReply to this comment

can any one please tell from where to prepare GENERAL AWARENESS for IBPS PO??atleast tell name of books or certain website..lot of data is available on internet but whom to trust…..really frustrated..

PrAdEePReply to this comment

good one


7 of 7 20-Jul-13 9:00 AM

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