Math Is Fun Forum

mathattack

Member

Registered: 2016-03-07

Posts: 17

Trig and Sequences

Hi!  I'm having quite a bit of trouble with this problem, could someone help?  I really want to know how to solve it.  I set up the triangle, but can't figure how to use the tan and cot to form geometric or arithmetic progressions.  Thanks!

In triangle ABC, AB = BC, and BD is an altitude. Point E is on the extension of AC such that BE = 10. The values of tan CBE, tan DBE, and tan ABE form a geometric progression, and the values of cot DBE, cot CBE, and cot DBC form an arithmetic progression. What is the area of triangle ABC?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Trig and Sequences

"cot DBE, cot CBE, and cot DBC form an arithmetic progression."
Please check this whether it is reproduced properly because cotangent is a decreasing function in the first quadrant
Correction on May 28:Forget about this comment. It only means angle DBC is smaller than angle CBE.

Last edited by thickhead (2016-05-27 16:03:49)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Trig and Sequences

Last edited by thickhead (2016-05-28 04:45:49)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Trig and Sequences

I'm working on an accurate diagram for this.  I have a method that ought to work but the algebra is horrendous. 

More will follow.

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

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Trig and Sequences

Are we supposed to take absolute values of tangents (ignoring the sign ) for geometric progression?

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Trig and Sequences

hi thickhead,

I'm using the following:

Sketchpad to make drawing. 
tan and cot as calculated by this software.
If A, B and C are in GP then A/B - B/C = zero
If D, E and F are in AP then D -2E + F = zero.

I then constructed a diagram and measured as necessary.  I calculated as above.  Then I used trial and improvement to juggle the points into the best positions.  Here is my best effort:

I cannot improve on these values because the minimum point movement is one pixel and any further movement moves me away from the optimum values.

My algebraic approach is to use the above formulas with distances DB, DC, DE etc substituted in.  CBE, and ABE are not in right angled triangles but I have been able to eliminate them using compound angle formulas for tan.  I also have (Pythag) BE^2 = 100 = (DB+DE)(BD-DE) so, in theory I can eliminate DE as well.

That would leave an equation only containing DB and DC, so I was hoping to reduce that to the required area.  Unfortunately the expression is too complicated (five lines of my sheet of A4) so I've stuck at that.

Tried to find simple formulas connecting the tan and cot expressions but no luck there either.

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

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Trig and Sequences

I went through the statement of the problem again. I had taken AB=AC but it is AB=BC. I have to work again.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Trig and Sequences

Last edited by thickhead (2016-06-01 05:32:49)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Trig and Sequences

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}