Once upon a time, when I was deep into trigonometry, I followed the trail of sine and cosine sums, with sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
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 –
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
Now the vertical from B is sin(2x)
so sin(2x)/(2sin(x)) is the ratio which should be cos(x)
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?”
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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 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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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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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.
How many points are there in 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.
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Trump, Watergate, Nixon, Rockefeller: the real lesson —Two very different men, two very different presidents, Trump and Nixon; but the real reasons for attacking them are the same— by J…
Source: Trump, Watergate, Nixon, Rockefeller: the real lesson
(Copied from elsewhere; author unknown but reprinted with thanks to whoever it was. Also thanks to my friend Kathryn Wadsworth who put this out on FB!) A DAY IN THE LIFE OF JOE REPUBLICAN AMERICAN …
Source: What have those godless anti-American libtards done for me anyway?
Now for more exiting and less mathematical stuff.
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
Have fun. The words are easy.
The row operation matrix inversion method is so neat and ingenious, and it has the same operations for all dimensions of matrix.
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.
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.
This is the standard way of inverting a matrix. I came across this first in 1960. Oh so long ago! Goodbye Cramer’s Rule.