88 lines
4.1 KiB
Markdown
88 lines
4.1 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
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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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| $sin(0) = 0$ | $cos(0) = 1$ | $tan(0) = \frac{cos(0)}{sin(0)} = \frac{0}{1} =0$ |
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| ----------------------------------------- | ----------------------------------------- | ---------------------------------------------------------------- |
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| $sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $tan(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2} = 1$ |
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| $sin(\frac{\pi}{2}) = 1$ | $cos(\frac{\pi}{2}) = 0$ | $tan(\frac{\pi}{2}) = \frac{1}{0} = DNF$ |
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Interpreting the above table:
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- When $x = 0$, $y = 0$
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- When $x = \frac{\pi}{4}$, $y = 1$
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- When $x = \frac{\pi}{2}$, there's an asymptote
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Without any transformations applied, the period of $tan(x) = \pi$. Because $tan$ is an odd function, $tan(-x) = -tan(x)$.
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# Cotangent
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$$ y = cot(x) $$
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To find relative points to create the above graph, you can use the unit circle:
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If $cot(x) = \frac{cos(x)}{sin(x)}$, then:
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| $sin(0) = 0$ | $cos(0) = 1$ | $cot(0) = \frac{sin(0)}{cos(0)} = \frac{1}{0} = DNF$ |
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| ----------------------------------------- | ----------------------------------------- | ---------------------------------------------------------------- |
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| $sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $cot(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2} = 1$ |
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| $sin(\frac{\pi}{2}) = 1$ | $cos(\frac{\pi}{2}) = 0$ | $tan(\frac{\pi}{2}) = \frac{1}{0} = DNF$ |
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Without any transformations applied, the period of $cot(x) = \pi$. Because $cot$ is an odd function, $cot(-x) = -cot(x)$.
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# Features of Tangent and Cotangent
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Given the form $y = A\tan(Bx - C) + D$ (the same applies for $\cot$)
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- The stretching factor is $|A|$
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- The period is $\frac{\pi}{|B|}$
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- The domain of $tan$ is all of $x$, where $x \ne \frac{C}{B} + \frac{\pi}{2} + {\pi}{|B|}k$, where $k$ is an integer. (everywhere but the asymptotes)
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- The domain of $cot$ is all of $x$, where $x \ne \frac{C}{B} + \frac{\pi}{|B|}k$, where $k$ is an integer (everywhere but the asymptotes)
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- The range of both is $(-\infty, \infty)$
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- The phase shift is $\frac{C}{B}$
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- The vertical shift is $D$
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# Examples
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> Given $-2\tan(\pi*x + \pi) - 1$
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$A = -2$, $B = \pi$, $C = -\pi$, $D = -1$
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| Transformation | Equation |
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| -------------- | ------------------------- |
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| Stretch | $\|-2\| = 2$ |
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| Period | $\frac{\pi}{\|\pi\|} = 1$ |
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| Phase shift | $\frac{-\pi}{\pi} = -1$ |
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| Vertical shift | $-1$ |
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