Then two more:

sin(A) + sin(B) = 2sin((A + B)/2)cos((A – B)/2)

and 2sin(A)cos(B) = sin(A + B) + sin(A – B)

Being unable to remember these last formulae (there are 8 altogether) I learned the basic one and derived, over and over again, the others.

In all of this I was never able to derive the basic formula until de Moivre’s theorem appeared.

So I had another go with trig and the simple version, and this is the result:

Aim of the game : sin(2x) = 2sin(x)cos(x)

First diagram:This does not look promising !

But what about sensible labelling –

Second diagram:

The vertical from F meets AD in J

and the line from F at right angles to AB meets AB in K.

So we have two pairs of congruent triangles, BH and HD are both equal to sin(x),

and BHD is A STRAIGHT LINE

Final diagram:

Now the vertical from B is sin(2x)

so sin(2x)/(2sin(x)) is the ratio which should be cos(x)

Fix it

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MiniEd: the Ministry of Education, Airstrip One, Oceania It was a warm but overcast day in late August and the clocks were striking thirteen. Mr Winston Smith, Principal of the Victory G+MINDSET Academy (formerly…]]>

2084 ?

No. Much closer.

MiniEd: the Ministry of Education, Airstrip One, Oceania

It was a warm but overcast day in late August and the clocks were striking thirteen.

Mr Winston Smith, Principal of the Victory G+MINDSET Academy (formerly the Bogstannard Comprehensive School), woke to find himself lying on something that felt like a camp bed, except that it was higher off the ground and it seemed that he was fixed down in some way so that he could not move. Light that seemed stronger than usual was falling on his face.

He gasped as he realised that the infamous MiniEd interrogator, “Grammar School” O’Greening, was standing at his side, looking down at him intently. At the other side of him stood a man in a white coat, tapping an iPad.

“Tell me, Winston,” said O’Greening gently, but with a chilling undercurrent of steel in her voice, “how many buckets am I holding up?”

Winston…

View original post 449 more words

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We are in dire straits with an all-out assault on science, evidence and reason coming out of the White House all day long, every day now. The direct attack on science is part…]]>

Not so optional !

**We are in dire straits** with an all-out assault on science, evidence and reason coming out of the White House all day long, every day now. The direct attack on science is part of a much-wider trend.

Science is imperfect and the truth has no billionaire backer fighting for it. The deck is stacked in the other direction.

Fear Uncertainty Doubt:Fear, uncertainty and doubt (often shortened to FUD) is a disinformation strategy used in sales, marketing, public relations, talk radio, politics, religious organizations, and propaganda. FUD is generally a strategy to influence perception by disseminating negative and dubious or false information and a manifestation of the appeal to fear.

…By spreading questionable information about the drawbacks of less well known products, an established company can discourage decision-makers from choosing those products over its own, regardless of the relative technical merits.

President FUD is the living…

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Wouldn’t it be wonderful if Democrats and Republicans set aside their differences & came together under one roof & then, due to decades of neglectful infrastructure funding policy, erosion collapsed that…]]>

Renamed “A bridge too far”.

A Narcissist Writes Letters, To Himself

Wouldn’t it be wonderful

if Democrats and Republicans

set aside their differences

& came together under one roof

& then, due to decades

of neglectful infrastructure funding policy,

erosion collapsed that roof?

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The Cantor Set is constructed in the following way: Start with the interval [0,1]. Next, remove the open middle third interval, which gives you two line segments [0,1/3] and [2/3,1]. Again, remove the middle third for…]]>

And they call them “real” numbers!

The Cantor Set is constructed in the following way:

Start with the interval [0,1]. Next, remove the open middle third interval, which gives you two line segments [0,1/3] and [2/3,1]. Again, remove the middle third for each remaining interval, which leaves you now with 4 intervals. Repeat this final step *ad infinitum*.

The points in [0,1] that do not eventually get removed in the procedure form the **Cantor set**.

Consider the diagram below:

An interval from each step has been coloured in red, and each red interval (apart from the top one) lies underneath another red interval. This nested sequence shrinks down to a point, which is contained in every one of the red intervals, and hence is a member of the Cantor set. In fact, each point in the Cantor set corresponds to a unique infinite sequence of nested intervals.

View original post 222 more words

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Source: Trump, Watergate, Nixon, Rockefeller: the real lesson

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Source: What have those godless anti-American libtards done for me anyway?

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I don’t know if this is “the way forward” or “just bullshit”, but McKinsey knows (something).

So on with “Digital” in education:

This is the introduction –

**Companies today are rushing** headlong to become more digital. But what does digital really mean?

*For some executives, it’s about technology. For others, digital is a new way of engaging with customers. And for others still, it represents an entirely new way of doing business. None of these definitions is necessarily incorrect. But such diverse perspectives often trip up leadership teams because they reflect a lack of alignment and common vision about where the business needs to go. This often results in piecemeal initiatives or misguided efforts that lead to missed opportunities, sluggish performance, or false starts.*

And the rest of it is here

http://www.mckinsey.com/industries/high-tech/our-insights/what-digital-really-means

Have fun. The words are easy.

;

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Here is a step by step approach, where firstly the dimension is chosen, then the first of the buttons is selected (Start). After which the buttons are selected in order. The states of the left and right matrices are displayed at each stage, and finally the identity matrix appears on the left, and the inverse of the original matrix appears on the right.

http://mathcomesalive.com/mathsite/matrix%20prog%203.html

The first display shows the original matrix on the left. Nothing has been done yet.

The second one is the 2 x 2 matrix inverted.

The third is the 4 x 4 matrix inverted.

The matrices can be altered with the file name for the application, “matrix prog 3.html”. using right click and “view page source”. You are then on your own, with javascript !!!!!!!!

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