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IIT JEE Trigonometric Maximum 2010 IIT JEE Paper 1 Problem 48 Trigonometric Maximum
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- the maximum value of the expression one over sine squared of theta plus three sine theta
- co-sign theta
- plus five cosine squared of theta is
- let's rewrite this, so this is one over
- one over, so i have a sine squared of theta
- i have a sine squared of theta and then
- at least my brain whenever i see a sine squared of theta i always look for cosine squared of theta because
- i know when i take the sum of them it equals one
- i don't have just one cosine squared of theta here at five cosine squared of thetas let me just
- take one of them
- so i have a plus cosine the squared theta
- because i took one of them i only have four of these left
- so plus four cosine squared of theta
- and then i have this stuff plus
- three sine
- of theta of co-sine of theta
- so what this
- first first step allowed me to do is just turn
- two characters right over here sine squared theta plus cosine squared theta
- that is equal to one
- so we've simplified it to one over
- one over
- one plus
- now lets think about how we can write co-signed squared of theta
- i'll write our identities here
- co-signed squared of theta we've proved these in the
- trigonometry playlist. This is equal to
- one-plus cosine
- one-plus co-signed of two theta
- all of that
- over
- two and my goal here is that i really
- just wanna get everything, well i just wanna simplify and maybe
- we will do calculus
- value of the numerator
- solicit cosine squared theta is equal to this
- so four times this is just going to be
- so four times this before divide by two instances would be two times this numerator
- so it's going to be too
- plus too
- co-signed
- of to data that's this term right here
- and then this term
- this term right over here we could use the tree guided me that
- signed of to data
- is equal to two sign of state a
- co-sign if data
- or you divide both sides right to you get a one-half sign of
- tuesday dot
- is equal to sign of feder
- co-sign of theta so this is this right over here this part right over here is going to
- be one-half signed up to date a bore multiplying it by three
- just going to be plus three halves
- sign of two data
- sighing of
- too
- glitzy thus
- part over here clearly simplifies
- this is the recent history right this is one over
- three
- plus
- to co-signed of to data
- plus thirty halves
- sign
- of to data
- now we really just looking for in the minimum
- we're looking for the
- minimum value of the denominator which would give us the same as the maximum value of the
- new murder or just be one over at this minimum values alexia how how lol we can get assuming
- were above zero t how low we can get
- for this denominator write your words will look for its minimum value so one day we can
- do just to simplify things
- the middle value of this
- is going to be the same demand
- the bid of this thing over right there
- to kind of confuses the problem
- the minimum value
- the replies to cosign
- to stayed up plus three have sign
- of two data
- is going to be the same thing is the minimum value of three plus
- on the screen to the substitution the tuesday it is equal to exodus simplify things a little
- bit you don't have to do that
- so three plus two co-signed of acts
- plus three halves
- sayed
- of acts
- so this is a pretty simple expression to see how we can figure out some minimum value and
- my temptation is to take the derivative find out where the derivative is equal to zero
- and then that will either be a maximum or a minimum point
- so let's take the derivatives the derivative of this expression right over here with respect
- to acs
- would really love three with the for texas zeroed route of two cosine of axis negative
- negative to sign of x
- derivative of three have side of actions will be plus
- three halves cosine of x
- and that is going to be equal to zero we want to find where the slope is your committee
- worry about somewhere minimum point
- and let's see we can
- admit
- we had to sign of extra boats side so we get the rehabs
- co-signed of acs is equal to two
- signed of x and then we can divide both the sides both sides of this equation
- by
- will survive like to first alaska too many steps of three
- forts co-signed of x
- is equal to sign
- of acs
- and to divide both sides by cosine of x
- so we get three
- over for is equal to sign of x over a co-signer of x
- which is the same thing as the pentagon that cs
- so uh... annexed by you that gives us three or attended of annexed by his three-fourths
- is going to give us either a maximum or a minimum point celeste think about this
- let's think about this a little bit will be drawn by unit circle
- so think about that too
- technip to expel use that will give us a tangent of three-fourths
- so that we draw my unit circle
- that's the unit circle
- redraw unit
- download this is always the hardest part so redraw this
- r ight
- their inspired units or call
- so how can i get the triangle
- or or
- we're listing but that would how can i get a triangle who where and angle is attended
- of three four two river candid is opposite over adjacent
- right tangent is opposite over adjacent so
- if missus wire triangle right over here
- if this is acs
- opposite over adjacent is equal to three fourths
- so opposite could be three an adjacent could be
- and we hopefully immediately recognize this this is the
- this is a three four five triangle too right triangle
- three square plus four square it is twenty five
- which is five squares as of three four
- triangle
- there's too
- headed values so x could be like this and this obviously isn't
- isn't eight unit high partners who over here but we can divide everything by five it would
- be so we could have the situation
- we could have been situation over here
- where this is actually this is the uh... circle the hypothesis one
- this is three over five
- and this is for over five
- this word tended of ax here
- canada backs would give us the reports but it was going to give us a maximum value were
- minimal value
- well over here both close sign
- both co-signed of acts and sign of x are going to be positive so both of these are going
- to be positive values so screw them
- probably maximize
- maximize the six pressure over here
- by the other
- x that gives us the same tangent
- remember the standard is really just the slope
- the radius of the unit circle would be
- the angle would be the single this is the same this has the same kenton value
- so dispatched
- this x
- over
- this actually and this case the tended to still
- the tended to still going to be three-fourths
- but over here
- the science co-signed our negatives over here
- overhear the
- xc ward at or the co-sign is going to be negative four fifths
- and the sign value
- or or the wide values used to be negative three-fifths
- and this will give us
- this will give us
- minimum boy because here everything is every ball the sign in the coastline are negative
- so let's use
- let's use this x right over here
- and notice we don't even have to figure out what the axis 'cause we know
- but if it ended overhears four-fifths
- the speed of both the sba science going to be
- the size of the three fifths of the course i was going to be forecast
- or which will give some actual point or the tenant could be
- for three-fourths and then the side will be negative benefits and of course i will be
- negative forfeit so let's use these overuse of the middle of
- is going to be equal to three plus
- went two times coastline of x_ were using this one over here so to transpose side of
- exco sign of experience
- negative four fists
- negative
- for fests
- and then plus three halves
- plus three has times in the sign of x sinai lectures negative three fifths
- negative
- three fits and what is this
- going to be equal to this is going to be equal to
- three
- plus
- this is negative eight fifths
- dreamweaver should write three minus eight fifth
- three minus the this
- minus nine tenths
- minus nine tenths and so this is going to be equal to
- we could put everything over ten
- thirty over ten
- minus sixteen over ten
- right that's eight fists
- minus nine or ten and this gives us what
- this gives us five or ten
- five over ten or or one-half
- so the minimum value the minimum value for denominator everything we've been dealing
- with so far has been our denominator
- the middle of the idea of all of this business over here
- the minimum value is one-half
- so the maximum value
- that this whole expression takes is when the minimum values one out so we get one over
- one-half
- equal to to we're done
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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