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vault/Mathematics/Algebra/Inequalities/Logarithmic Inequalities.md
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| >[!DEFINITION] Definition: Logarithmic Inequality | ||
| > | ||
| >A **(real) logarithmic inequality** is an [inequality](Inequality.md) which contains variables as part of the base or the argument of a logarithm. | ||
| > | ||
| >[!ALGORITHM]- Algorithm: Solving Inequalities of the Form $\log_a f(x) \gt \log_a g(x)$ | ||
| > | ||
| >We are given the following inequality: | ||
| > | ||
| >$$ | ||
| >\log_a f(x) \gt \log_a g(x) | ||
| >$$ | ||
| > | ||
| >Solutions: | ||
| >- If $0 \lt a \lt 1$, then $\begin{cases}f(x) \lt g(x) \\ f(x) \gt 0\end{cases}$ | ||
| >- If $a \gt 1$, then $\begin{cases}f(x) \gt g(x) \\ g(x) \gt 0\end{cases}$ | ||
| > | ||
| >[!ALGORITHM]- Algorithm: Solving Inequalities of the Form $\log_{b(x)}f(x) \ge \log_{b(x)} g(x)$ | ||
| > | ||
| >We are given the following inequality: | ||
| > | ||
| >$$ | ||
| >\log_{b(x)}f(x) \ge \log_{b(x)} g(x) | ||
| >$$ | ||
| > | ||
| >Solutions: | ||
| > | ||
| >$$ | ||
| >\begin{cases}0 \lt b(x) \lt 1 \\ f(x) \le g(x) \\ f(x) \gt 0\end{cases} \qquad \bigcup \qquad \begin{cases}b(x) \gt 1 \\ f(x) \ge g(x) \\ g(x) \gt 0 \end{cases} | ||
| >$$ | ||
| > |
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...equalities/Trigonometric Inequalities/Fundamental Trigonometric Inequalities.md
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| >[!ALGORITHM] Algorithm: Solving Inequalities of the Form $\sin x \lt c, \sin x \le c, \sin x \ge c, \sin x \gt c$ | ||
| > | ||
| >>[!ALGORITHM] Algorithm: Solving Inequalities of the Form $\sin x \lt c$ or $\sin x \le c$ | ||
| >> | ||
| >>We are given a [trigonometric inequality](Trigonometric%20Inequality.md) of the following form. | ||
| >> | ||
| >>$$ | ||
| >>\sin x \lt c | ||
| >>$$ | ||
| >> | ||
| >>Solutions: | ||
| >>- If $c \le -1$, then $x \in \varnothing$. | ||
| >>- | ||
| >> | ||
| >[!ALGORITHM]- Algorithm: Solving Inequalities of the Form $\sin x \lt c$ and $\sin x \le c$ | ||
| > | ||
| >We are given a [trigonometric inequality](Trigonometric%20Inequality.md) of the following form. | ||
| > | ||
| >$$ | ||
| >\sin x \lt c | ||
| >$$ | ||
| > | ||
| >Solutions: | ||
| > | ||
| >$$ | ||
| >x \in \bigcup_{k \in \mathbb{Z} \left(-\pi - \alpha_0 + 2k\pi; \alpha_0 + 2k\pi \right) | ||
| >$$ | ||
| > | ||
| >[!ALGORITHM]- Algorithm: Solving Inequalities of the Form $\cos x \lt c$ and $\cos x \le c$ | ||
| > | ||
| >We are given a [trigonometric inequality](Trigonometric%20Inequality.md) of the following form. | ||
| > | ||
| >$$ | ||
| >\cos x \lt c | ||
| >$$ | ||
| > | ||
| >Solutions: | ||
| >- If $c \le -1$, then $x \in \varnothing$. | ||
| > | ||
| >[!ALGORITHM]- Algorithm: Solving Inequalities of the Form $\tan x \gt c$ and $\tan x \ge c$ | ||
| > | ||
| >We are given a [trigonometric inequality](Trigonometric%20Inequality.md) of the following form. | ||
| > | ||
| >$$ | ||
| >\tan x \gt c | ||
| >$$ | ||
| > | ||
| >Solutions: | ||
| > | ||
| >$$ | ||
| >x \in \bigcup_{k \in \mathbb{Z}} \left(\alpha_0 + k\pi; \frac{\pi}{2}+k\pi \right) | ||
| >$$ | ||
| > | ||
| >[!ALGORITHM]- Algorithm: Solving Inequalities of the Form $\cot x \lt c$ and $\cot x \le c$ | ||
| > | ||
| >We are given a [trigonometric inequality](Trigonometric%20Inequality.md) of the following form. | ||
| > | ||
| >$$ | ||
| >\cot x \lt c | ||
| >$$ | ||
| > | ||
| >Solutions: | ||
| >- If $c \le -1$, then $x \in \varnothing$. | ||
| >- | ||
| > |
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...ate Systems of Euclidean Space/Coordinate Transformations in Euclidean Space.md
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| >[!THEOREM]- Theorem: Cartesian $\leftrightarrow$ Polar | ||
| > | ||
| >If $\mathbf{p} \in \mathbb{R}^2$ has [Cartesian coordinates](Cartesian%20Coordinate%20System.md) $(x,y)$, then its [polar coordinates](Polar%20Coordinate%20System.md) $(r, \theta)$ are: | ||
| > | ||
| >- Using the convention $r \ge 0$ and $\theta \in (-\pi; \pi]$: | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| >r &= \sqrt{x^2 + y^2} \\ | ||
| > | ||
| >\theta &= | ||
| > | ||
| >\begin{cases} | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) \qquad \qquad \text{if } x \gt 0 \\ | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) + \pi \qquad \text{ if } x \lt 0, y \ge 0 \\ | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) - \pi \qquad \text{ if } x \lt 0, y \lt 0 \\ | ||
| > | ||
| >\frac{\pi}{2} \qquad \qquad \qquad \qquad \text{if } x = 0, y \gt 0 \\ | ||
| > | ||
| >-\frac{\pi}{2} \qquad \qquad \qquad \, \, \, \, \, \, \text{ if } x = 0, y \lt 0 \\ | ||
| > | ||
| >0 \qquad \qquad \qquad \qquad \, \text{ if } x = y = 0 | ||
| > | ||
| >\end{cases} | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >- Using the convention $r \ge 0$ and $\theta \in [0; 2\pi)$: | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >r &= \sqrt{x^2 + y^2} \\ | ||
| > | ||
| >\theta &= | ||
| > | ||
| >\begin{cases} | ||
| > | ||
| >\arccos \frac{x}{\sqrt{x^2 + y^2}} \qquad \qquad \text{ if } y > 0 \\ | ||
| > | ||
| >2\pi - \arccos \frac{x}{\sqrt{x^2 + y^2}} \qquad \text{if } y < 0 \\ | ||
| > | ||
| >0 \qquad \qquad \qquad \qquad \hphantom{,,,,} \text{if } y = 0, x \ge 0 \\ | ||
| > | ||
| >\pi \qquad \qquad \qquad \qquad \hphantom{,,,,} \text{if } y = 0, x \lt 0 | ||
| > | ||
| >\end{cases} | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >If $\mathbf{p} \in \mathbb{R}^2$ has [polar coordinates](Polar%20Coordinate%20System.md) $(r, \theta)$, then its [Cartesian coordinates](Cartesian%20Coordinate%20System.md) $(x,y)$ are | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >x &= r \cos \theta \\ | ||
| > | ||
| >y &= r \sin \theta | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >>[!PROOF]- | ||
| >> | ||
| >>TODO | ||
| >> | ||
| > | ||
| >[!THEOREM]- Theorem: Cartesian $\leftrightarrow$ Spherical | ||
| > | ||
| >If $\mathbf{p} \in \mathbb{R}^3$ has [Cartesian coordinates](Cartesian%20Coordinate%20System.md) $(x,y,z)$, then its [spherical coordinates](Spherical%20Coordinate%20System.md) $(r, \theta, \phi)$ are: | ||
| > | ||
| >- Using the convention $r \ge 0$, $\theta \in [0; \pi]$ and $\phi \in (-\pi; \pi]$: | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >r &= \sqrt{x^2 + y^2 + z^2} \\ | ||
| > | ||
| >\theta &= | ||
| >\begin{cases} | ||
| > | ||
| >0 \qquad \qquad \qquad \hphantom{,,,,,} \text{if } x = y = z = 0 \\ | ||
| > | ||
| >\arccos \frac{z}{\sqrt{x^2 + y^2 + z^2}} \hphantom{,,,,} \text{otherwise } | ||
| > | ||
| >\end{cases} \\ | ||
| > | ||
| >\phi &= | ||
| > | ||
| >\begin{cases} | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) \qquad \qquad \text{if } x \gt 0 \\ | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) + \pi \qquad \text{ if } x \lt 0, y \ge 0 \\ | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) - \pi \qquad \text{ if } x \lt 0, y \lt 0 \\ | ||
| > | ||
| >\frac{\pi}{2} \qquad \qquad \qquad \qquad \text{if } x = 0, y \gt 0 \\ | ||
| > | ||
| >-\frac{\pi}{2} \qquad \qquad \qquad \, \, \, \, \, \, \text{ if } x = 0, y \lt 0 \\ | ||
| > | ||
| >0 \qquad \qquad \qquad \qquad \, \text{ if } x = y = 0 | ||
| > | ||
| >\end{cases} | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >- Using the convention $r \ge 0$, $\theta \in [0; \pi]$ and $\phi \in [0; 2\pi)$: | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >r &= \sqrt{x^2 + y^2 + z^2} \\ | ||
| > | ||
| >\theta &= | ||
| >\begin{cases} | ||
| > | ||
| >0 \qquad \qquad \qquad \hphantom{,,,,,} \text{if } x = y = z = 0 \\ | ||
| > | ||
| >\arccos \frac{z}{\sqrt{x^2 + y^2 + z^2}} \hphantom{,,,,} \text{otherwise } | ||
| > | ||
| >\end{cases} \\ | ||
| > | ||
| >\phi &= | ||
| > | ||
| >\begin{cases} | ||
| > | ||
| >\arccos \frac{x}{\sqrt{x^2 + y^2}} \qquad \qquad \text{ if } y > 0 \\ | ||
| > | ||
| >2\pi - \arccos \frac{x}{\sqrt{x^2 + y^2}} \qquad \text{if } y < 0 \\ | ||
| > | ||
| >0 \qquad \qquad \qquad \qquad \hphantom{,,,,} \text{if } y = 0, x \ge 0 \\ | ||
| > | ||
| >\pi \qquad \qquad \qquad \qquad \hphantom{,,,,} \text{if } y = 0, x \lt 0 | ||
| > | ||
| >\end{cases} | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >If $\mathbf{p} \in \mathbb{R}^3$ has [spherical coordinates](Spherical%20Coordinate%20System.md) $(r, \theta, \phi)$, then its [Cartesian coordinates](Cartesian%20Coordinate%20System.md) $(x,y,z)$ are | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >x &= r \sin \theta \cos \phi \\ | ||
| > | ||
| >y &= r \sin \theta \sin \phi \\ | ||
| > | ||
| >z &= r \cos \theta | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >>[!PROOF]- | ||
| >> | ||
| >>TODO | ||
| >> | ||
| > | ||
| >[!THEOREM]- Theorem: Cartesian $\leftrightarrow$ Cylindrical | ||
| > | ||
| >If $\mathbf{p} \in \mathbb{R}^3$ has [Cartesian coordinates](Cartesian%20Coordinate%20System.md) $(x,y,z)$, then its [Cylindrical coordinates](Cylindrical%20Coordinate%20System.md) $(\rho, \phi, z)$ are: | ||
| > | ||
| >- Using the convention $\rho \ge 0$ and $\phi \in (-\pi; \pi]$: | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >\rho &= \sqrt{x^2 + y^2} \\ | ||
| > | ||
| >\phi &= | ||
| > | ||
| >\begin{cases} | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) \qquad \qquad \text{if } x \gt 0 \\ | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) + \pi \qquad \text{ if } x \lt 0, y \ge 0 \\ | ||
| > | ||
| >\arctan \left( \frac{y}{x} \right) - \pi \qquad \text{ if } x \lt 0, y \lt 0 \\ | ||
| > | ||
| >\frac{\pi}{2} \qquad \qquad \qquad \qquad \text{if } x = 0, y \gt 0 \\ | ||
| > | ||
| >-\frac{\pi}{2} \qquad \qquad \qquad \, \, \, \, \, \, \text{ if } x = 0, y \lt 0 \\ | ||
| > | ||
| >0 \qquad \qquad \qquad \qquad \, \text{ if } x = y = 0 | ||
| > | ||
| >\end{cases} \\ | ||
| > | ||
| >z &= z | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >- Using the convention $\rho \ge 0$ and $\phi \in [0; 2\pi)$: | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >\rho &= \sqrt{x^2 + y^2} \\ | ||
| > | ||
| >\phi &= | ||
| > | ||
| >\begin{cases} | ||
| > | ||
| >\arccos \frac{x}{\sqrt{x^2 + y^2}} \qquad \qquad \text{ if } y > 0 \\ | ||
| > | ||
| >2\pi - \arccos \frac{x}{\sqrt{x^2 + y^2}} \qquad \text{if } y < 0 \\ | ||
| > | ||
| >0 \qquad \qquad \qquad \qquad \hphantom{,,,,} \text{if } y = 0, x \ge 0 \\ | ||
| > | ||
| >\pi \qquad \qquad \qquad \qquad \hphantom{,,,,} \text{if } y = 0, x \lt 0 | ||
| > | ||
| >\end{cases} \\ | ||
| > | ||
| >z &= z | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >If $\mathbf{p} \in \mathbb{R}^3$ has [Cylindrical coordinates](Cylindrical%20Coordinate%20System.md) $(\rho, \phi, z)$, then its [Cartesian coordinates](Cartesian%20Coordinate%20System.md) $(x,y,z)$ are | ||
| > | ||
| >$$ | ||
| >\begin{align*} | ||
| > | ||
| >x &= \rho \cos \phi \\ | ||
| > | ||
| >y &= \rho \sin \phi \\ | ||
| > | ||
| >z &= z | ||
| > | ||
| >\end{align*} | ||
| >$$ | ||
| > | ||
| >>[!PROOF]- | ||
| >> | ||
| >>TODO | ||
| >> | ||
| > |
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