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Trignometry

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14 Aug 2011 12:33:28 IST

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solve the equation

Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Trigonometry

e^{cos x}=e^{-cos x}+ 4

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Yagyadutt Mishra

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14 Aug 2011 14:07:21 IST

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Hello !
e^{cos x}=e^{-cos x}+ 4
Let e^{cos x} = t

Then t = t^-1 + 4
t = 1/t + 4
t^2 = 1 + 4t
t^2 -4t - 1 = 0

t = 2 + root(5) and t = 2 - root(5)

As t = e^{cos x} and cosx has a maximum value of 1 ..so maximum value of t must be equal to t = e which is approximately 2.733 something ...and minimum value of t = 1/e which is in fraction and positive ...
So t = 2+root(5) must not be the answer ..because it become greater than three ...(not allowed because tmax = 2.733)
And t = 2-root(5) is neglected because it become negative .....and we know tmin is positive ...!

So the given equation has zero solution :)

G.Divya Sweta

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15 Aug 2011 19:29:11 IST

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hi... but will u explain y u have used cap after t in 4th step.... nd plz. explain the logic behind it...

G.Divya Sweta

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15 Aug 2011 19:32:50 IST

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nd i also have problem regarding finding the solutions in trigonometry... hw to solve it?

Yagyadutt Mishra

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Joined: 19 Feb 2009

Posts: 1979

15 Aug 2011 22:11:12 IST

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*cap* is used for power notation ..!

t^2 means ...t square
t^-1 means ... t to the power -1 ..

Since i have assume e^cosx as t so e^-cosx will be equal to (e^cosx )^-1 = t^-1

And the substitute these values in equation again ...   t = t^-1 + 4    and then the solution ..(I am not going to solve this again in the same step)

Logic is ...The assumption ...whenever u feel there can be a quadratic equation after assuming some constant ..go for it...I am highlighting few examples ...

1) sinx^2 + 5cosx + 3 = 0

Here u can assume cosx = t => 1-cos^2x + 5cosx + 3 = 1-t^2 + 5t + 3 = 0
or t^2 -5t - 4 = 0   ( and then further u can solve it ) and after getting the value of t suppose u get ..

t=0.5 and t = -3    ...so in this case t=-3 will be neglected because ...as we have assumed t equals to cosx ..which means t can not be less than -1 and greater 1 ...so definitely t=-3 ..doesn't fall in the range ..!

Hence the acceptable answer is t=0.5 ..

2) ln x + 2/lnx = 4

Now how can u solve this without substituting ....Surely assume lnx = t

t + 2/t = 4   again a quadratic equation and ....blah blah blah..(the same step as i have posted above)

3) 2^cosx + 2^-cosx = 5

Again ....say 2^cosx = t ...and further the same step ..!

I hope it is clear to u now ..

nd i also have problem regarding finding the solutions in trigonometry... hw to solve it?

The only way to master yourself in trigonometry is hard practice ...go for examples ...first ..solve as many as question as u can ..in begining u will find difficult to solve it..don't worry ...solve the same question again and again if u will something illogical in it....don't finish unless u understand the logic and steps completely ..

A.Das Gupta for trigonometry is i guess the best book covering and giving a handful knowledge about the topic...Further when u feel confidence ..Go for standard book like M.Khanna ...and Arihant ..and do some tougher problems too ..!

Good luck

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