The Ungerecht Family Tree

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=====Trigonometry=====

transformations of sine or cosine function: 
 y = a * sin(b(x - c)) + d
 
{{:urp:sinewave.jpg?418|Sine Wave Transformations}}

(Graphics from www.Desmos.com, labeling was added) a = amplitude, c = phase shift, d = vertical shift, 2π/b = period
(for tan the period is π/b)

==== Sin/Cos table ====

^  Radians    ^  Degrees  ^  Cos  ^  Sin  ^
|   **0**     |   0      |   1    |   0   |
|   π/6       |   30     |  √3/2  |  1/2  |
|   π/4       |   45     |  √2/2  |  √2/2  |
|   π/3       |   60     |  1/2  |  √3/2  |
|   **π/2**   |   90     |   0   |   1   |
|   2π/3      |   120    |  -1/2  |  √3/2  |
|   3π/4      |   135    |  -√2/2  |  √2/2  |
|   5π/6      |   150    |  -√3/2  |  1/2  |
|   **π**       |   180    |  -1   |   0   |
|  -5π/6, 7π/6  |   210    |  -√3/2  |  -1/2  |
|  -3π/4, 5π/4  |   225    |  -√2/2  |  -√2/2  |
|  -2π/3, 4π/3  |   240    |  -1/2  |  -√3/2  |
|  **-π/2, 3π/2**  |   270    |   0    |  -1   |
|  -π/3, 5π/3   |   300    |  1/2   |  -√3/2  |
|  -π/4, 7π/4   |   315    |  √2/2  |  -√2/2  |
|  -π/6, 11π/6  |   330    |  √3/2  |  -1/2  |
|  **0, 2π**      |   360    |   1   |   0   |

==== Conversion Theorems ====
double angle: 2θ
  sin(2θ) = 2sin(θ)cos(θ)
  cos(2θ) = cos²(θ) - sin²(θ)
          = 2*cos²(θ) - 1
          = 1 - 2*sin²(θ)

negative angles:
  sin(-θ) = -sin(θ)
  cos(-θ) = cos(θ)
  tan(-θ) = -tan(θ)

additive, subtractive:
  sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
  cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
  sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
  cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

complements:
  sin(θ) = cos(π/2 - θ)
  cos(θ) = sin(π/2 - θ)

=== See also ===
  * https://www2.clarku.edu/faculty/djoyce/trig/identities.html
  * 11 pages of definitions, postulates and theorems: http://www.ouchihs.org/ourpages/auto/2013/7/26/52822673/Geo-PostulatesTheorems-List.pdf

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The Ungerecht Family Tree

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