Math Is Fun Forum

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Trig

Hi,

Could anyone help with this:

Show that:

can be expressed as:

Thanks,
Mitch.

Last edited by gnitsuk (2006-08-29 01:38:28)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Trig

Mitch, did you ever get an answer for this?

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igloo myrtilles fourmis

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Re: Trig

Hi,

No, I never got a reply to this or the subsequent one. Apart from a triple angle substitution and lots of algebra, I have no other ideas. The context of the problem is in a caculus text. I have arrived at the first formula and the answer given is in the second form. Given that it's a calculus question I would be suprised if one had to do lots af algebra at the end just to get into the form given - still, it's possible. Moreover. I was looking for a quicker method which I had missed.

Thanks,
Mitch.

Last edited by gnitsuk (2006-09-03 21:57:08)

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Trig

I've got some progress, and I wish you can put the finishing touch.

Firstly, notice 1-cos(2t)=2sin[sup]2[/sup](t)

Thus you get the chance of cancelling sin[sup]2[/sup](t) in the denominator and the numerator together.

And the crucial step is to express the denominator as some product with sin[sup]2[/sup](t).

the second and the last part are ok, but what about the first one?

use a[sup]3[/sup]-b[sup]3[/sup]=...

sin(3t)-sin(t)=2sin(t)cos(2t) -- one sin(t)

sin(3t)^2=?

Use Euler formula and find sin(3t) expressed as trig of t. In this way you can complete another sin(t)

sin(3t)=3cos[sup]2[/sup](t)sin(t)-sin[sup]3[/sup](t)

Euler Formula

Last edited by George,Y (2006-10-01 01:04:09)

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X'(y-Xβ)=0

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Re: Trig

Hi, thanks for your help.

I finally did it this way:

Result (1):

therefore

So from:

we can replace

in the numerator with

and

in the denominator with our result (1) to get:

Now divide numerator and denominator by

to give

Equation (1):

Result (2)

Result (3)

Now we can substitute Results (2) and (3) into the denominator of Equation (1) to give:

Divide numerator and denominator by 8 giving:

Call this Equation (2)

Result (4):

Substitute this into equation (2) to finally give:

Last edited by gnitsuk (2006-10-02 03:21:30)