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- Trig-Identities
- Min-Max table
- Ratta-fication formulas
- The AM GM Logic
- Find minimum value of 4 tan2θ + 9 cot2θ
- Extra facts
- SSC CGL 2012 Tier II Question
- The least value of 2 sin2 θ + 3 cos2 θ (CGL2012T1)
- The maximum value of Sin x + cos x is
- The maximum value of 3 Sin x – 4 Cos x is
- Min Max values of sin 4x + 5 are
- 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 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
- a sin θ ± b cos θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }
- a sin θ ± b sin θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }
- a cos θ ± b cos θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }
- 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
- 3
- 5
- 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
- 2
- 3
- 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
- √2
- 1/ √2
- 1
- 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
- 5
- 7
- 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
- 2, 6
- 4, 5
- -4, -5
- 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
- Do not exist
- -1, 1
- Sin -1 , Sin +1
- – 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
min value of [2+sin(a)]/[2+cos(a)]
why min(4sec^2x+cos^2x) not equal to 4
By A.M.>=G.M., we get it 4.
But actually it is 5.
Explain why?
why min(4sec^2x+cos^2x) is not equal to 4
By A.M.>=G.M., it is 4
But actually it is 5.
By Double Differenciation its answer is 5
Can you explain why
3sin^2x + 4sinxcosx + 5cos^2x
Sir can u please tell me the minimum value of this?
QUITE USEFUL
thanku very much mrunal sir
refer ur solved ex. sin^2 θ + cos^2 θ + sec^2 θ + cosec^2 θ + tan^2 θ + cot^2 θ =7. This can be solved through AM-GM logic.
sin^2 θ + cos^2 θ + sec^2 θ + cosec^2 θ + tan^2 θ + cot^2 θ = ( Sin^2θ+Cosec^2θ)+(Cos^2θ+Sec^2θ)+(Tan^2θ+Cot^2θ) =2x{Root (Sin^2θxCosec^2θ)} + 2x{Root (Cos^2θxSec^2θ)} + 2x{Root(Tan^2θxCot^2θ) }
=2x(Root1) +2x(Root1) + 2x(Root1) = 3x2x1=6.
Which answer is correct? Please suggest me.
=(sin^2x+Cos^2x)+(sec^2x-Tan^2x)+(cosec^x-Sec^2x)+Tan^2x+Tan^2x+Cot^2x+Cot^2x (Adding +Tan^2x & -Tan^2x)
=3+2(Tan^2x+Cot^2x)
=3+2((Tanx-cotx)^2+2)
As the square value is always +ve the maximum value is 3+(2*2)=7
Abhijit… dis is my query too…plz rply to me if u hav got ur answer
Sir, in Ssc cgl tier 2 2012 stion : if we find minimum value of Sin^2x + cos^2x + sec^2x + cosec^2x + tan^2x+ cot^2x …. cannot we apply AM GM concept in all of dem instead of doing sin^2x + cos^2 x =1 ….. and if we are doing so, den the answer is coming out “6” … P.S : THANX A TON FOR POSTING THIS, IT WAS REALLT HELPFUL … PLZ REPLY
Sir please solve this
A= sin^2@+cos^4@ find minimum value
minimum of (tahn^2x-cot^2x) solve it plz
minimum of (tahn^2x-cot^2x) solve it plzz
-2
-4 is minimum value
break it into sin and cos and then take L.C.M u will get something (sin^4x -cos^4x)/(cosxsinx)^2 = ( (sin^2x +cos^2x)(sin^2x-cos^2x))/(cosxsinx)^2 = 1(cos2x)/(1/4) = 1*-1/(1/4) (min.value of cos2x=-1 and that of cosxsinx=-1/2) =-4 ans
so,ur answer is -4
(sin^2x +cos^2x)(sin^2x-cos^2x))/(cosxsinx)^2 = 1.(1-cos^2x-cos^2x)/(1/4)=4(1-2cos^2x)=4-4*0=+4(min. value of cos^2nx = 0)
so why the min value not be as +4 instead of -4.
(sin^2x-cos^2x)/sin^2xcos^2x= -cos2x/(1/4)= -4cos2x hear we should substitute min value of cos2x such that function value should be minimum so we substitute min value of cos2x as +1 insted of -1
if -1 is substituted ans is +4
if +1 is substituted ans is -4
ultimately we need min value so we sub +4, so the min value is -4
Pls tell me the minimum value of 1/sin^2cos^x
If tan x equals n*tan y ,then find the maximum value of tan square(x-y) in terms of n.
How to find minimum value of sin^2x+cos^4x using above concepts?
sin^2x+ cos^2x(cos^2x)
=sin^2x+ cos^2x(1-sin^2x)
=sin^2x +cos^2x -cos^2x sin^2x
=1- (sinxcosx)^2
min value : 1- (1/2)^2 =3/4 (for minimum we put max value of sinxcosx)
max value : 1-0 =1 (for max we put min value of sinxcosx due to -ve sign in the statement)
In the above explanation, it is given like maximum value of sinxcosx is +1/2 and min value is -1/2…..U took the min value of sinxcosx as ‘ 0 ‘…Why..???
Hi,
If we take minimum value it would become positive due to square term. so in order to minimize the entire function it should be taken as 0.
u rock bro
hello mrunal sir this article is nice but
THERE IS A BIG PROBLEM WITH THIS QUESTION WHICH U HAVE EXPLAINED IN APPROACH #1
2sin²@ + 3cos²@
ur answer is 2
but in rajesh verma’s fast arithmetic page no 691 its answer is 3
now who is correct as the logic by rajesh verma is also correct plsssss help me out I am preparing for SSC cgl :-(
How come is Rajesh Verma correct? It is obvious that 2sin²@ + 3cos²@(min.) = 2(sin²@ + cos²@) + cos²@ = 2+0 = 2
what is the max and min value of 2^sinx + 2^cosx
Respected sir, please solve this problem, find the minimum value of 4sec(power2)theta plus 9 cosec(power2)theta
16
Answer is correct min value is 2. And max value is 3
hi
If Min. value of (sin θ cos θ)n = (½)n
Then minimum value of sinxcosx should be 1/2 right? But it is giv3n that its min value is -1/2
It is (-1/2)^n
Sir in the question – “What is the min. value of Sin^2θ + Cos^2θ + Sec^2θ + Cosec^2θ + Tan^2θ + Cot^2θ ?”, If we make three pairs as (Sin^2θ+Cosec^2θ)+(Cos^2θ+Sec^2θ)+(Tan^2θ+Cot^2θ), then all the three pairs gives min. value=2, by usiing A.M ≥ G.M logic. And the overall answer becomes 6. So please tell what’s wrong in this approach?
here we can’t conclude the minimum value of separate part as we are to find out minimum value of entire function.
as in case of (Sinx +cosx) if we go for individual approach the answer would be -1-1 =-2 but if we go for entire expression the answer would be underroot(1+1) = Underroot(2).
So we are supposed to use identities as much as possible like sin^2 x + cos^2 x = 1 and turn the expression at last stage so that we can put min or max value as per requirement.. Hope it clears.
Visual. Actually when applying am.gm inequality in we have to check for the defining of function like no.must be positive and defined for some particular value of x and most important that range of function must be continuous like of tan and cot which is R. so we cannot apply am.gm for cosec^2x + sin^2x etc.
trick is convert function to tan and cot square and make for sin and cos different and then solve.
bahut hi acha post hai sir…thanks
hai…catch me..09059411838
The maximum value of sin^4 θ+cos^4 θ is.
A.1 B.2 C.3 D.1/3
Sir please tell
Is answer 2 ?
sin^ x + cos^4 x = (sin^2 x +cos^2 x)^2 – 2 (sinx. cosx)^2
= 1 – 2 (0) as in order to have maximum value of expression we have to keep sin x. cosx = 0
PLEASE TELL ME THE MAX. VALUE OF SIN^4 THETA + COS^4 THETA
1 aayega agr kabi bhi sin cos ki power even ho tho max value 1 hoti hai
1/2
Can we apply AM>=GM logic to
Find out minimum value of 4sin^2 a+9 cos^ 2a?
no
Find the maximum and minimum value of sin^4x+cos^2x and also find the maximum value of sin^1000x+cos^2008x
Sir in the question – “What is the min. value of Sin^2θ + Cos^2θ + Sec^2θ + Cosec^2θ + Tan^2θ + Cot^2θ ?”, If we make three pairs as (Sin^2θ+Cosec^2θ)+(Cos^2θ+Sec^2θ)+(Tan^2θ+Cot^2θ), then all the three pairs gives min. value=2, by usiing A.M ≥ G.M logic. And the overall answer becomes 6. So please tell what’s wrong in this approach?
this method is wrong coz AM>GM always but for the equality to hold all term must but equal for at least one value of Theta for the pairs you have made no value of theta exist for which simultaneously squares of all Trig ratios are equal so
thank you mrunaal
I’m very grateful to you… Thank you so much.
And the emoji you used in between gave the feel of live lecture :)
Find the minimum value of Cos^2x+cos^2-cos^z