Math Is Fun Forum

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

trigonometry#3

If

prove that:-

Last edited by niharika_kumar (2014-04-18 16:36:18)

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friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: trigonometry#3

hi Niharika,

Put tan = sin/cos and square the equation.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: trigonometry#3

I did that but couldn't succeed in proving the above.

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friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: trigonometry#3

hi Niharika,

I'm sorry; I thought this would come out easily.  Changing the tan is the standard thing I would do for one like this.  Then everything is in terms of sine and cos so it should have worked.

But I cannot do it either.  Below I have shown the graphs for (i) tan and root(sin - cos) and then for (ii) sin and cos cubed. 

The first pair only cross in positive real numbers and twice in each 2pi repeat of the graphs.  The second pair cross once only in positive values in each 2pi repeat.  And nowhere near the same value of x (theta).  So it would seem that this cannot be shown for real numbers.  Odd.

Is it possible there's a 'typo' somewhere?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: trigonometry#3

The exact question is :-
If θ≠n*pi and tanθ is the geometric mean between sinθ and cosθ , then prove that 2-4sin^2θ+3sin^4θ-sin^6θ=1

geometric mean was given what I mentioned and the solution solved the above in just a single line saying

after that sinθ was taken in terms of cos^3θ and the question was solved easily.

But I couldn't get how this result was obtained..

Last edited by niharika_kumar (2014-04-20 02:31:36)

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friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: trigonometry#3

I think the geometric mean of a and b is

so for your question

Now what I suggested will work. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: trigonometry#3

hi Niharika can you please use the latex code for your expression cause the question seems quite clear and i did get sinθ=cos^3 θ
use the G.M. formula if tanθ is the G.M. between sinθ and cosθ then tanθ=sqroot(sinθ*cosθ) then u easily get sinθ=cos^3 θ so just post the expression clearly so we can work it out

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: trigonometry#3

http://www.mathwords.com/g/geometric_mean.htm

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: trigonometry#3

may be the sign was put in by mistake.
thanks for the correction, the answer came out easily.:D
I spent 2 hrs around it and wasted so many sheets of paper, phew!!

Last edited by niharika_kumar (2014-04-20 03:03:37)

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friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: trigonometry#3

After half an hour I start suspecting an error and make the graphs. 

Especially as I make sign mistakes all the time. 

2-4sin^2θ+3sin^4θ-sin^6θ=1

Did you manage this bit?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: trigonometry#3

yes.
I did put the equation as

then I used the formula

and simplified it.

hence it was proved.

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friendship is tan 90°.