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urp:trig
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urp:trig [2021-10-15] nerf_herder |
urp:trig [2021-10-21] 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 | + | |
| + | (Graphics from www.Desmos.com, labeling was added) a = amplitude, c = phase shift, d = vertical shift, 2π/k = period | ||
| + | for tan, the period is π/k | ||
| + | |||
| + | ==== 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) | ||
| - | double angle: 2theta | ||
| - | sin(2theta) = 2sin(theta)cos(theta) | ||
| - | cos(2theta) = cos^2(theta) - sin^2(theta) | ||
| - | = 2*cos^2(theta) - 1 | ||
| - | = 1 - 2*sin^2(theta) | ||
| negative angles: | negative angles: | ||
| - | sin(-theta) = -sin(theta) | + | sin(-θ) = -sin(θ) |
| - | cos(-theta) = cos(theta) => tan(-theta) = -tan(theta) | + | 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(theta) = cos(pi/2 - theta) | + | sin(θ) = cos(π/2 - θ) |
| - | cos(theta) = sin(pi/2 - theta) | + | cos(θ) = sin(π/2 - θ) |
| === See also === | === See also === | ||
urp/trig.txt · Last modified: 2021-10-21 by nerf_herder
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