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urp:trig [2021-10-15] nerf_herder |
urp:trig [2021-10-17] nerf_herder |
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Line 2: | Line 2: | ||
transformations of sine or cosine function: | transformations of sine or cosine function: | ||
- | y = a * sin(k(x + c)) + d | + | y = a * sin(k(x - c)) + d |
- | a = amplitude, -c = phase shift, d = vertical shift, 2pi/k = period | + | {{:urp:sinewave.jpg?418|Sine Wave Transformations}} |
- | for tan, the period is pi/k | + | |
- | double angle: 2theta | + | (Sine wave & graph courtesy of www.Desmos.com) a = amplitude, c = phase shift, d = vertical shift, 2π/k = period |
+ | for tan, the period is π/k | ||
+ | |||
+ | ---------- | ||
+ | double angle: 2θ | ||
sin(2θ) = 2sin(θ)cos(θ) | sin(2θ) = 2sin(θ)cos(θ) | ||
cos(2θ) = cos^2(θ) - sin^2(θ) | cos(2θ) = cos^2(θ) - sin^2(θ) | ||
- | = 2*cos^2(θ) - 1 | + | = 2*cos^2(θ) - 1 |
- | = 1 - 2*sin^2(θ) | + | = 1 - 2*sin^2(θ) |
negative angles: | negative angles: | ||
sin(-θ) = -sin(θ) | sin(-θ) = -sin(θ) | ||
- | cos(-θ) = cos(θ) => tan(-θ) = -tan(θ) | + | cos(-θ) = cos(θ) |
+ | tan(-θ) = -tan(θ) | ||
additive, subtractive: | additive, subtractive: | ||
- | sin(a+b) = sin(a)cos(b) + cos(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) | + | cos(a+b) = cos(a)cos(b) - sin(a)sin(b) |
- | sin(a-b) = sin(a)cos(b) - cos(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) | + | cos(a-b) = cos(a)cos(b) + sin(a)sin(b) |
complements: | complements: | ||
- | sin(θ) = cos(pi/2 - θ) | + | sin(θ) = cos(π/2 - θ) |
- | cos(θ) = sin(pi/2 - θ) | + | cos(θ) = sin(π/2 - θ) |
=== See also === | === See also === |