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**Titus****Member**- Registered: 2005-03-07
- Posts: 10

The structure below measures 10m in length by 0.8m in width. In total, there are 21 diagonal pieces exactly like AB and two extra pieces- CD. Pieces CD are parallel to pieces AB. Pieces AB are inclined at 60 degrees from the horizontal.

Please help mefind the length (in millimetres) of piece CD as i already know how to find the length of AB.

After that i need to find the total length of the out frame and all 21 pieces of AB.

Any help is greatly appreciated.

Note: the vertical lines (apart from those on the outer frame) are not a part of the actual structure. They are merely usaed to form triangles so certain lengths can be calculated.

thanks.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Making a right-angled triangle out of AB, the line going down from B and the line going right from A, we have one angle and one length. The 'opposite' side is 0.8m, the angle is 60° and we want to find the 'adjacent' side. tanθ=o/a, so tan 60=0.8/a, so a=0.8/tan 60=0.461... There are 21 of these, so 9.6995... of the 10m is being taken up by them. That leaves 0.3005... for CD. There are 2 of CD, so they will get 0.1503... each.

Making another right-angled triangle out of CD and the corner, we now know the side length and we want to know the hypotenuse. cosθ=a/h, so cos 60=0.1503.../h, so h=0.1503.../cos 60=0.30m to the nearest cm, which is length CD.

The outer frame has two lengths and two widths on it, so in total it is 10+10+0.8+0.8=21.6m long and one length AB is 0.8/sin 60=0.9238...m, so 21 of them are 19.40m to the nearest cm.

Why did the vector cross the road?

It wanted to be normal.

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**ahgua****Guest**

Solve this:

2sinx.cosx + cos^2x = 1

**wcy****Member**- Registered: 2005-08-04
- Posts: 117

1=cos^2 x+sin^2 x

then, group all the sin terms...

2 sinx cosx +cos^2x=cos^2x +sin^2 x

2 sinx cosx=sin^2 x

sin^2 x-2 sinx cos x=0

sin x(sin x-2 cos x)=0

sin x=0 or sin x=2 cos x

x=0,etc..

or tan x=2

x=...

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**ahgua****Guest**

wcy wrote:

sin x=2 cos x

tan x=2

x=...

Are you sure we can divide sin x by cos x?

what if cos x = 0 ...

then, the answer will be undefined!

**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,085

You are right!

When x=0,

2sinx.cosx + cos^2x = 1

But how do we get a solution x=0?

2sinx.cosx + cos^2x = 1

sin2x + cos^2x = 1,

sin2x = 1-cos^2x

sin2x = sin^2x

this would be true only when x=0

But is there any (other)mathematical way it can be shown that x=0 is a solution?

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ajp3****Member**- Registered: 2005-08-25
- Posts: 9

ahgua, do not worry about dividing by cos x in the (correct) sol'n given by wcy; sin x/cos x is defined as tan x, so we do not need to worry about division by zero, the dfn of tan x already forbids it!

another way to show x=0 is a sol'n was already given by wcy in that soln; the sin x = 0 half of the soln...

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**ahgua****Member**- Registered: 2005-08-24
- Posts: 25

thanks ajp3. But I got another question on trigo...

Given cosec A + cot A = 3

and thus evalute cosec A -cot A

and find cos A

*Last edited by ahgua (2005-08-25 22:52:31)*

Life is a passing dream, but the death that follows is eternal...

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,085

cosecA + cotA = 3

To find cosecA - cotA,

one should remember,

cosec²A - cot²A = 1

From this, we get

(cosecA+ cotA)(cosecA - cotA)=1

Given cosecA + cotA = 3,

3(cosecA-cotA)=1

cosecA-cotA=1/3

cosecA + cotA + cosecA-cotA = 3+1/3 = 4/3

2cosecA = 4/3

cosecA=2/3

sinA=3/2 funny

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ahgua****Member**- Registered: 2005-08-24
- Posts: 25

Prove that (sec^2 x + 2 tan x)/ (1 - tan^2 x) = (cos x + sin x)/(cos x - sin x)

Hence solve the equation (sin x +cos x)/(sin x - cos x) = tan x for 0 < x < 2pi.

Life is a passing dream, but the death that follows is eternal...

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