49 lines
1.6 KiB
Markdown
49 lines
1.6 KiB
Markdown
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# Sine/Cosine
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Given the above graph:
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- At the origin, $sin(x) = 0$ and $cos(x) = 1$
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- A full wavelength takes $2\pi$
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# Manipulation
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| Formula | Movement |
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| ---------------- | ---------------------------------- |
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| $y = cos(x) - 1$ | Vertical shift down by 1 |
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| $y = 2cos(x)$ | Vertical stretch by a factor of 2 |
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| $y = -cos(x)$ | Flip over x axis |
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| $y = cos(2x)$ | Horizontal shrink by a factor of 2 |
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# Periodic Functions
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A function is considered periodic if it repeats itself at even intervals, where each interval is a complete cycle, referred to as a *period*.
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# Sinusoidal Functions
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A function that has the same shape as a sine or cosine wave is known as a sinusoidal function.
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There are 4 general functions:
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| $$A * sin(B*x - C) + D$$ | $$ y = A * cos(B*x -c) + D$$ |
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| ----------------------------------------- | -------------------------------------- |
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| $$ y = A * sin(B(x - \frac{C}{B})) + D $$ | $$ y = A*cos(B(x - \frac{C}{B})) + D$$ |
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How to find the:
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- Amplitude: $|A|$
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- Period: $\frac{2\pi}{B}$
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- Phase shift: $\frac{C}{|B|}$
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- Vertical shift: $D$
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$$ y = A * \sin(B(x-\frac{C}{B})) $$
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# Tangent/Cotangent
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$$ y = tan(x) $$
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To find relative points to create the above graph, you can use the unit circle:
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If $tan(x) = \frac{sin(x)}{cos(x})$, then:
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| $sin(0)$ | | |
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| -------- | --- | --- |
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$$ y = cot(x) $$
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