Submitted by emoticons in lobby

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Have unit square sides of 1. make Rotation it with angle λ.

Unit circle radius 1 with origin center is has point Q on the squares corner. is [ cos λ , sin λ ]

an Other Unit circle radius 1 but Q at centers of it. has point P on the square corner. horizontal line thru Q will do angle λ + [π / 2] between P and Q and the line

P point equaling [ cos λ + cos[λ + [π / 2]] , sin λ + sin[λ + [π / 2]] ]

now Unit circle radius √2 with origin centered is will also has P. but angle λ + [π / 4]

P point equaling [ √2 cos[λ + [π / 4]] , √2 sin[λ + [π / 4]] ]

now and cos λ + cos[λ + [π / 2]] = √2 cos[λ + [π / 4]]

now and sin λ + sin[λ + [π / 2]] = √2 sin[λ + [π / 4]]

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moonlune wrote

I tried a different approach for fun.

cos λ + cos[λ + [π / 2]] = √2 cos[λ + [π / 4]]
sin λ + sin[λ + [π / 2]] = √2 sin[λ + [π / 4]]

this system is just the real and imaginary part of the same complex:

exp(iλ) + exp(i(λ+π/2)) = √2 exp(i(λ+π/4))

which can be written:

exp(iλ) [1 + exp(iπ/2)] = exp(iλ) √2 exp(iπ/4)
[1 + exp(iπ/2)] = √2 exp(iπ/4)
1 + i = √2 (√2/2 + i√2/2)
1 + i =  1 + i

QED.

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emoticons OP wrote

Good one! cool Perspectif solving the 2 equation system for see each λ is a solve :)

QED.

Q E Deez nuts gottem lol /lh

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Tecate_Coyote wrote (edited )

Magic wizard with the magic words.

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