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A sine wave has four properties, namely amplitude, period, phase shift and vertical shift.
Amplitude is half the distance between the maximum point and the minimum point. It determines the vertical stretch of the sine wave, i.e. the higher the amplitude, the taller the sine wave, and the lower the amplitude, the shorter the sine wave. In terms of the unit circle, amplitude is the radius of the unit circle, namely 1.
Period is the horizontal distance needed to complete one oscillation cycle. It determines the horizontal stretch of the sine wave, i.e. the shorter the period, the more compressed the sine wave, and the longer the period, the more stretched the sine wave.
Angular frequency is closely related to period. It tells us how many complete oscillation cycles the sine wave does every 2π radians (360°). Because the sine function is based on the unit circle, angular frequency is usually measured as the number of complete oscillation cycles every 2π radians. The angular frequency of the base sine wave is 1 as f(θ) = sin θ completes exactly one full oscillation cycle over the interval [0, 2π]. Furthermore, angular frequency is inversely proportional to the period T, namely f = 2π / T. This means the higher the frequency, the shorter the period, and the lower the frequency, the longer the period.
The phase shift is how far the sine wave is shifted to the left or to the right from its original position.
The vertical shift is how far the sine wave is shifted up or down from its original position, while midline is the horizontal line that cuts the sine wave into two halves.
The general form of the sine wave is defined by the following equation:
f(θ) = A sin(Bθ + C) + D, where
i) A is the amplitude
ii) B is the angular frequency
iii) C is the phase shift
iv) D is the vertical shift
The general form of the sine wave allows us to add transformations. Specifically, by changing the values of the variables A and B in the general form of the sine wave, we manipulate the shape of the sine wave; and by changing the values of the variables C and D, we manipulate the position of the sine wave. For the base sine wave (i.e., f(θ) = sin θ), A = 1, B = 1, C = 0 and D = 0.
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00:00 Patreon ad
00:10 Intro
00:27 Graphing the base sine wave, i.e. f(θ) = sin θ
00:59 Manipulating the amplitude of the sine wave
02:05 Manipulating the angular frequency of the sine wave
03:13 Manipulating the phase shift of the sine wave
04:22 Manipulating the vertical shift of the sine wave
05:42 Multiple transformations added to a sine wave
08:11 Support my work on social media
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You can get the Python script that generates the animation from my Patreon page, namely / sohcahtoa1609 .
Playlists
1) Geometry: • Geometry
2) Trigonometry: • Trigonometry
3) Arithmetic: • Arithmetic
4) Algebra: • Algebra
5) Differential calculus: • Differential calculus
6) Conic sections: • Conic sections (geometry)
7) Triangles: • Triangles (geometry)
8) Quadrilaterals: • Quadrilaterals (geometry)
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