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A simple rule for the simple sines
It is an often-used fact (in examples of elementary trigonometry and problems that involve it) that the sines of certain simple angles have simple expressions themselves. Specifically,
| sin(0) | = | 0 | ||
| sin(π/6) | = | sin(30°) | = | 1/2 |
| sin(π/4) | = | sin(45°) | = | √(2)/2 |
| sin(π/3) | = | sin(60°) | = | √(3)/2 |
| sin(π/2) | = | sin(90°) | = | 1 |
Furthermore, these statements about angles in the first quadrant can be reflected to handle similar common angles in the other three quadrants, and cosines. There is a common diagram illustrating this, considering sine and cosine as the coordinates of points on the unit circle.
When attempting to memorize these facts as one is expected to, I observed a pattern: each of the values may be expressed in the form √(x)/2.
| sin(0) | = | √(0)/2 | ||
| sin(π/6) | = | sin(30°) | = | √(1)/2 |
| sin(π/4) | = | sin(45°) | = | √(2)/2 |
| sin(π/3) | = | sin(60°) | = | √(3)/2 |
| sin(π/2) | = | sin(90°) | = | √(4)/2 |
I found this pattern to be quite useful. It does not, however, explain why those particular angles (quarters, sixths, eighths, and twelfths — but not tenths! — of circles) form this pattern of sines.
no subject
The relationships between the π/6, π/4 pair and the π/3, π/2 pair are of course a consequence of the half-angle or double-angle identity — sin 2θ = 2 sin θ cos θ — and the relationship between sin and cos, namely cos θ = √(̅1̅ ̅-̅ ̅s̅i̅n̅²̅ ̅θ̅)̅. Substituting, sin 2θ = 2 sin θ √(̅1̅ ̅-̅ ̅s̅i̅n̅²̅ ̅θ̅)̅. In the case where θ = π/4 and sin θ = √2̄/2, we have sin 2θ = 2 √2̄/2 √(̅1̅ ̅-̅ ̅(̄√̄2̄̄/̄2̄)̅²̅)̅ = 2 √2 √(̅1̅-̅2̅/̅4̅)̅ = √2̅ √½̅ = √2̅ √2̅/2 = 2/2 = 1. I don't know that this sheds any real light on the topic, though...
This Unicode abuse is brought to you by my compose key (http://canonical.org/~kragen/setting-up-keyboard.html) and my keyboard map (https://github.com/kragen/xcompose).
no subject
I've done similar things with my Mac OS X keyboard layout; I haven't yet gotten around to posting it though. Rather than a single Compose key I use Alt/Option keys to shift into several alternate sets: Greek, Logic, Set, Superscript/Subscript, Arrows. Compose would probably be nicer, but it didn't occur to me to do something like that (and I would probably have to allocate a non-modifier key to it).
For a teaser, here's my documentation file (the un-mentioned keystrokes are the same as the default Mac US keyboard layout)
Ordinary/shift key: Swapped [] and (), swapped : and ;, swapped | and \. Option key: 3 × times symbol 4 ¢ cents symbol (Standard) 9 · center dot symbol (usually shifted) 0 ‚ low-9 quotation mark (usually shifted) d ∂ partial derivative symbol (Standard) t þ Thorn (lowercase) h θ Theta (lowercase) -- TODO: Decide if having theta on Greek is good enough j ∆ U+2206 Increment (Standard - capital delta operator symbol) a Dead: Arrows WASD are directions, QEZC are diagonals, RF are double arrows Shift doubles the bar. s Dead: Set symbols u ∪ Union i ∩ Intersection e ∈ Element of 0 ∅ Empty set , ⊂ Subset . ⊃ Superset k Dead: Greek keyboard Same as system Greek layout. Note that Option-w is summation sign ∑ whereas Option-k Shift-s is capital sigma Σ. l Dead: Logical symbols A ∀ For all E ∃ There exists a ∨ Or e ∧ And q ≡ Equivalence o ⇔ Biconditional i ∈ Element of (Option-i not-element-of) n ¬ Logical not x ≈ Proportional to (Standard) v √ Square root sign (Standard) b ∫ Integral sign (Standard) m µ mu sign (Standard) Option-Shift: ` ∝ Proportional-to symbol 6 Dead: superscripts and subscripts Superscripts supported: 1243567890-=+()in Press Option to access subscripts. Subscripts supported: 1234567890-=+()aeox t Þ Thorn (uppercase) f ′ Prime g ″ Double prime h † Dagger j ∇ Nabla (inverted delta) l λ Lambdano subject