Chapter 0: Mathematical Fundamentals

0.1 Function operations

0.1.1 Trigonometric Functions

secx and cscx

range：$(-\infty,-1]\cup[1,+\infty)$
Value: Calculate using the reciprocals of $\cos x$ and $\sin x$
Image:

Inverse Trigonometric Functions

Concept:
Note:
The inverse sine function is not the inverse function of $y=\sin x$ across the entire domain; note the range of $y=\arctan x$;
It is merely the inverse function of $y=\sin x$ on $\left[-\frac\pi2,\frac\pi2\right]$; solving inverse functions over other intervals requires the identity transformation to translate onto $\left[-\frac\pi2,\frac\pi2\right]$ to be represented. The inverse cosine function is similar;
Relationship: arcsinx and sinx
At $\left[-\frac\pi2,\frac\pi2\right]$, from the inverse function $f^{-1}[f(x)]=x$, we know: $arc\sin(\sin x)=x$
At $\left[-\frac\pi2,\frac\pi2\right]$, from the inverse function $f[f^{-1}(x)]=x$, we know: $\sin(\arcsin x)\equiv x$
Example: Using the identity transformation to translate x:
Find $\arcsin (\sin x),x\in(\frac{3}{4}\pi,\pi)$
Use the identity $\sin x=\sin(\pi-x)$, which moves the angle into $(0,\frac{\pi}{4})$, where the inverse-function identity applies.
Get $$\arcsin (\sin x)=\arcsin(\sin(\pi-x))=\pi-x, x\in (\frac{3}{4}\pi,\pi)$$
arcsinx：
range：$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$
arccosx：
range：$[0,\pi]$

Supplement: Definition of $\arctan$

$\tan(\arctan x)= x$
Illustration
Table

Basics of Trigonometric Functions

Sum and Difference Formula:
Formula:
$$\begin{aligned}&\sin\left(x\pm y\right)=\sin x\cos y\pm\cos x\sin y\\&\cos\left(x\pm y\right)=\cos x\cos y\mp\sin x\sin y\\&\tan\left(x\pm y\right)=\frac{\tan x\pm\tan y}{1\mp\tan x\tan y}.\end{aligned}$$
Common conclusions:
$$\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)$$
$$\sin(n\pi+\alpha)=(-1)^n\sin\alpha$$
Induction formula:
Reciprocal relationship:
$$\tan\alpha\cot\alpha=1、\sin\alpha\csc\alpha=1、\cos\alpha\sec\alpha=1$$
Quotient relationship:
$$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}、\cot\alpha=\frac{\cos\alpha}{\sin\alpha}$$
Square relation:
$$ \sin^{2}\alpha+\cos^{2}\alpha=1、1+\tan^{2}\alpha=\sec^{2}\alpha、1+\cot^{2}\alpha=\csc^{2}\alpha$$
Double angle formula:
$$\sin2\alpha=2\sin\alpha\cos\alpha$$
$$\cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$$
$$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}=\frac{2\cot\alpha}{\cot^2\alpha-1}=\frac{2}{\cot\alpha-\tan\alpha}$$
half-angle formula:

\begin{aligned} &\sin{\frac{\alpha}{2}}=\pm{\sqrt{\frac{1-\cos\alpha}{2}}} \\ &\cos{\frac{\alpha}{2}}=\pm\sqrt{\frac{1+\cos\alpha}{2}} \\ &\tan{\frac{\alpha}{2}}={\frac{\sin\alpha}{1+\cos\alpha}}={\frac{1-\cos\alpha}{\sin\alpha}}=\pm{\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}} \\ &\cot{\frac{\alpha}{2}}={\frac{1+\cos\alpha}{\sin\alpha}}={\frac{\sin\alpha}{1-\cos\alpha}}=\pm{\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}} \end{aligned}

Demotion and promotion
Lowering formula:
$$\cos^{2}\alpha=\frac{1+\cos2\alpha}{2},\quad\sin^{2}\alpha=\frac{1-\cos2\alpha}{2},\quad\tan^2\alpha=\frac{1-\cos2\alpha}{1+\cos2\alpha}$$
Promotion formula:
$$1+\cos2\alpha=2\cos^2\alpha,1-\cos2\alpha=2\sin^2\alpha, 1\pm\sin2\alpha=\left(\sin\alpha\pm\cos\alpha\right)^2$$
$\cos x$ $1$ Characteristics:
Example: $$1-\cos x=1-(\cos 2*\frac{x}{2})=1-(1-2\sin^2 \frac{x}{2})=2\sin^2 \frac{x}{2}$$
Inverse trigonometric function:
If $x=a\sin t$, then $t=\arcsin\frac{x}{a}$.
If $x=a\sec t$, then $t=\arccos\frac{a}{x}$.

Collection of Trigonometric Differentiations

Sine and Cosine:
$$(\sin x)^{\prime}=\cos x、(\cos x)^{\prime}=-\sin x$$
tan、cot、sec、csc：
$$\begin{aligned}(\tan x)^{\prime}&=\sec^2x&(\cot x)^{\prime}&=-\csc^2x\\\\(\sec x)^{\prime}&=\sec x\tan x&(\csc x)^{\prime}&=-\csc x\cot x\end{aligned}$$
Inverse trigonometric function:
$$\begin{aligned}&(\arcsin x)^{\prime}=\frac1{\sqrt{1-x^2}}&&(\arccos x)^{\prime}=-\frac1{\sqrt{1-x^2}}\\&(\arctan x)^{\prime}=\frac1{1+x^2}&&(\operatorname{arccot}x)^{\prime}=-\frac1{1+x^2}\end{aligned}$$

0.1.2 Polar Coordinates

Basic Concepts of Polar Coordinates

Transformation:
$$\left\{\begin{array}{c}x=r\cos\theta\\y=r\sin\theta\end{array}\right.\Leftrightarrow\left\{\begin{array}{c}r=\sqrt{x^2+y^2}\\\theta=\operatorname{atan2}(y,x)\end{array}\right.$$
Note: The above conversion method is limited to the case where the two origins are the same, and the polar coordinates reference system and the $x$ axis direction of the Cartesian coordinates are the same;

Common Forms of Polar Coordinates Functions

Ray y=kx, its polar coordinate equation: $\theta=\theta_0$
A circle with center at the origin and radius a: $r=a$
A circle with center (a,0) and radius a: $r=2a\cos\theta$
A circle with center (0,a) and radius a: $r=2a\sin\theta$

0.1.3 Parametric Equations

Parametric Equations

Straight line:
Format:
$y=k(x-x_{0})+y_{0}$
Parametric Equations:
$\begin{cases}x=x_0+t\cos\alpha\\y=y_0+t\sin\alpha\end{cases}$
Illustration:
Circle:
Format:
$(x-a)^2+(y-b)^2=r^2$
Parametric Equations:
$\begin{cases}x=a+r\cos t\\y=b+r\sin t\end{cases}$
Illustration:

Common Parametric Equations in Elementary Mathematics

\begin{aligned} &\text{Circle: } \left\{\begin{array}{l}x=r\cos t\\y=r\sin t\end{array}\right. \\ &\text{Ellipse: }\begin{cases}x=a\cos t\\y=b\sin t&\end{cases} \\ &\text{Hyperbola: }\left\{\begin{array}{l}x=a\sec t\\y=b\tan t\end{array}\right. \\ &\text{Parabola: }\left\{\begin{array}{l}x=2ct\\y=t^2\end{array}\right. \\ &\text{spiral: } \left\{\begin{matrix}x=t\cos lt\\y=t\sin lt\end{matrix}\right. \\ &\text{Cycloid: } \left\{\begin{array}{l}x=r\cdot(t-\sin t)\\y=r\cdot(1-\cos t)\end{array}\right. \end{aligned}

Common Parametric Equations and Their Graphs

Star line
General formula $$x^{\frac23}+y^{\frac23}=a^{\frac23}$$
Parametric equations $$\begin{cases}x=a\cos^3\theta\\y=a\sin^3\theta\end{cases}$$
Image:
Cycloid
$$\text{ Parametric equations }\begin{cases}x=a (\theta-\sin\theta)\\y=a (1-\cos\theta)\end{cases}.$$
Logarithmic threads:
$$\rho=a\mathrm{e}^{\theta}$$
Double twisted wire

0.1.4 Logarithm and Exponent

Logarithmic Operation Rules

$$log_a(mn)=log_am+log_an$$
Note: $\log_2(m+n)$ cannot be split into a sum or product of $\log_2m$ and $\log_2n$; only products and quotients can be expanded using the rules above.
$$log_am^n=nlog_am$$
$$log_a\left(\frac{m}{n}\right)=log_am-log_an$$
$$log_a\sqrt[n]{m}=\frac{1}{n}log_am$$
$$\log_ab=\frac{\log_cb}{\log_ca}$$
$$u^v=\mathrm{e}^{v\ln u}\left(u>0\right)$$

Exponential Operation Rules

\begin{gathered} a^{0}=1\left(a\neq0\right) \quad\quad\quad\quad a^{r}\cdot a^{s}=a^{r+s} \\ (a^{r})^{s}=a^{rs} \quad\quad\quad\quad \frac{a^{r}}{a^{s}}=a^{r-s}(a\neq0) \\ \left(ab\right)^{r}=a^{r}b^{r} \quad\quad\quad\quad a^{-r}=\frac{1}{a^{r}}(a\neq0) \end{gathered}

Logarithmic and Exponential Operations

Take logarithms on both sides of the equation:
Function: Take logarithms at the same time, eliminate the exponents, and get the logarithms
$$y=(x+1)^{3}(x-2)^{2}\rightarrow lny=3ln(x+1)+2ln(x-2)$$
Taking exponents on both sides of the equation:
Function: At the same time, take the index, eliminate the logarithm, and get the index
$$y=\ln\left(x+{\sqrt{x^{2}+1}}\right)\rightarrow e^y=x+\sqrt{x^2+1}$$

0.1.5 Elementary Functions

Square, Cube, Multiple Formulas

The sum of the three terms:
$$\left(a+b+c\right)^2=a^2+b^2+c^2+2ab+2ac+2bc$$
Perfect cube sum and difference:
$$\begin{aligned}\left(a+b\right)^3&=a^3+b^3+3a^2b+3ab^2\\\left(a-b\right)^3&=a^3-b^3-3a^2b+3ab^2\end{aligned}$$
Binomial Theorem:
$$\begin{aligned}&\left(a+b\right)^{n}=\sum_{k=0}^{n}\mathrm{C}_{n}^{k}a^{n-k}b^{k}=a^n+\mathrm{C}_n^1a^{n-1}b+\cdots+\mathrm{C}_n^{n-1}ab^{n-1}+b^n\end{aligned}$$

Rationalizing Numerators and Denominators

Physics and chemistry -> Remove radicals from the numerator or denominator
numerators have physical chemistry:
Multiply the numerator and denominator by the conjugate radical of its numerator;
Example: $$f(x)=\frac{\sqrt{x+1}-\sqrt{x-1}}{x}=\frac{(\sqrt{x+1}-\sqrt{x-1})*(\sqrt{x+1}+\sqrt{x-1})}{x(\sqrt{x+1}+\sqrt{x-1})}=\frac{2}{x\cdot(\sqrt{x+1}+\sqrt{x-1})}$$
Here $\sqrt{x+1}+\sqrt{x-1}$ is the conjugate of $\sqrt{x+1}-\sqrt{x-1}$.
Rationalizing the denominator:
The method is the same as the rationalization of the numerator, multiply the numerator and denominator by the conjugate radical of its denominator;
Rationalizing both the numerator and denominator:
First multiply by the conjugate of the radical in the numerator, then by the conjugate of the radical in the denominator.

Partial-Fraction Decomposition

Method:
Step 1: Express the fraction as a sum of simpler fractions with unknown coefficients such as $A$ and $B$.
Step 2: Put the fractions over a common denominator and expand the numerator.
Step 3: Compare coefficients with the original numerator and solve for $A$ and $B$.
Example:
$$\text{Decompose }\quad\frac{x}{x^2-5x+6}.$$
$$\frac{x}{x^2-5x+6}=\frac{x}{(x-2)(x-3)},\qquad \text{let }\frac{x}{(x-2)(x-3)}=\frac{A}{x-2}+\frac{B}{x-3}.$$
Combining the fractions gives $$\frac{\text{A}(x-3)+\text{B}(x-2)}{(x-2)(x-3)}=\frac{(\text{A+B})x-3\text{A}-2\text{B}}{(x-2)(x-3)}.$$
Comparing coefficients gives $$\left.\left\{\begin{array}{c}\left(1\right)\mathrm{A}+\mathrm{B}=1\\\left(2\right)\mathrm{3A}+2\mathrm{B}=0\end{array}\right.\right.$$
Thus B=3 and A=-2, so $$\frac{x}{x^2-5x+6}=-\frac{2}{x-2}+\frac{3}{x-3}.$$
Example 2:

Add, Subtract, Split

Purpose:
Simplifying polynomial fractions;
Method:
Perform polynomial division from the highest-degree term downward until the degree of the remainder is less than the degree of the denominator.

Quadratic Equations in One Variable

To $ax^2+bx+c=0\left(a\neq0\right)$:
$\triangle = b^2- 4ac> 0\Leftrightarrow$ The equation has two distinct real roots;
$\triangle = b^2- 4ac= 0\Leftrightarrow$ The equation has two equal real roots;
$\bigtriangleup=b^2-4ac<0\Leftrightarrow$ The equation has a pair of conjugate roots
Root Formula:
$$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Solving the cubic equation

Step 1: Rewrite the equation in the form of $x^3+bx^2+cx+d=0$;
Step 2: Guess the roots, and based on d, guess which product of the three numbers it can be decomposed into;
Step 3: Organize and factorize -> first organize it into $(x-x_0)\left(x^2+px+q\right)=0$, solve $x^2+px+q=0$, and get two more roots;

Conclusion

arctangent function:
$$\arctan x+\arctan\frac1x\equiv\frac\pi2(x>0).$$
The following index is written directly:
$$(-1)^{2n}=1\quad\quad(-1)^{2n-1}=-1$$
As $n\to\infty$, $(-1)^{n-3}=(-1)^{n-1}$.
$$x^n-1=(x-1)(x^{n-1}+x^{n-2}+\cdots+x+1)$$

0.2 Geometric Foundations

0.2.1 Straight Line

Common Forms of Straight Lines

Point-slope form: $$y-y_0=k\left(x-x_0\right)$$
Two-point form: $$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\rightarrow \frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}.$$
Standard form: $$Ax+By+C=0$$

Formula for the distance from point to straight line

Distance formula:
$$d=\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$$

Normal, Tangent

Tangent equation: $y-y_{0}=f^{\prime}(x_{0})(x-x_{0})$
Normal equation: $y-y_0=-\frac1{f^{\prime}(x_0)}(x-x_0)$

0.2.2 Circles and Ellipses

Circles and Their Standard Forms

Standard form: $$(x-a)^2+(y-b)^2=R^2$$
Perimeter: $$2\pi R$$
Area: $$\pi R^2$$
Surface area of a sphere: $$S=4\pi R^2$$
Volume of a sphere: $$\frac{4}{3}\pi R^3$$

Volume of the Cone

S is the base area of the cone, H is the height of the cone, then:
$$V_{\text{cone}}=\frac{1}{3}SH$$
If the radius of the base of the cone is R, then:
$$V_{\text{cone}}=\frac{\pi}{3}R^2H$$

Ellipses and Their Areas

Elliptic equation: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Eccentric ellipse equation:
Shift m along the x-axis, shift n along the y-axis
$$\frac{\left(x-m\right)^2}{a^2}+\frac{\left(y-n\right)^2}{b^2}=1$$
Elliptical area: $$S=\pi ab$$

0.2.3 Area of a triangle

Concept: Two formulas for the area of a triangle

Let x, y, and z be the three side lengths, let y be the chosen base, let h be the corresponding height, and let the perimeter be 2p.
$$S=\sqrt{p(p-x)(p-y)(p-z)}=\frac12yh$$

0.2.4 Volume

Concept

Tetrahedral volume:
$$V_{\text{tetrahedron}}=\frac{1}{3}S_{\text{base}}h$$

0.3 Sequences and Mathematical Induction

0.3.1 Summing Sequences

Arithmetic and Geometric Sequences

Summation of arithmetic sequences: $$s_n=\frac{(a_1+a_n)n}{2}$$
Summing geometric sequences: $$s_n=\begin{cases}na_1,\quad(q=1),\\\frac{a_1(1-q^n)}{1-q},\quad(q\neq1).\end{cases}$$

Common Summation Conclusions

$$\sum_{k=1}^{1}\frac{1}{4k^{2}-1}=\frac{1}{2}\sum_{k=1}^{n}(\frac{1}{2k-1}-\frac{1}{2k+1})=\frac{1}{2}(1-\frac{1}{2k+1})$$
$$\sum_{k=1}^{n}k=\frac{1}{2}n(n+1)$$
$$\sum_{k=1}^{n}k^{2}=\frac{1}{6}n(n+1)(2n+1)$$

0.3.2 Mathematical Induction

First Mathematical Induction

The first mathematical induction method is suitable for propositions involving recurrence that only involve $a_1$ and $a_n$ terms, with the following steps:
(1) Verify that the proposition is correct when n=1 is valid;
Substitute n=1 into the left and right sides of the equation to see if it holds;
(2) Assume the proposition is correct when n=k;
Set up any number k, and assume the current k is n, then substitute it into the left and right sides of the expression;
(3) Prove that the proposition is correct at n=k+1
Take n=k+1, substitute it into the left side (summation expression) and the right side of the equation, and organize it into the form of an equation n=k+1;

Second Mathematical Induction

Example:
Let $D_n= 2aD_{n- 1}- a^{2}D_{n- 2}$, $D_{1}= 2a$, $D_2=3a^2\left(a\neq0\right)$, and use mathematical induction to prove: $D_n=\left(n+1\right)a^n.$
The second mathematical induction method is suitable for recursive propositions involving $a_{n-1}$, $a_n$, and $a_{n+1}$, with the following steps:
(1) Verify that the propositions at n=1 and n=2 are correct;
When $n=1$ $D_1=2a=\left(1+1\right)a^{1}$,
When $n=2$, $D_2= ( 2+ 1) a^2$, the proposition holds,
(2) Assume the proposition is correct when n<k;
Assume $n<k$ and that $D_n=(n+1)a^n$ holds.
(3) Proving that the proposition is correct when n=k is proven;
Then when n=k, $\begin{aligned}&D_k=2aD_{k-1}-a^2D_{k-2}=2a\left(ka^{k-1}\right)-a^2\left[\left(k-1\right)a^{k-2}\right]\end{aligned}=\begin{pmatrix}k+1\end{pmatrix}a^k.$
It can be determined by mathematical induction;

0.4 Other Knowledge

0.4.1 Complex Numbers

Concept

Complex number: for real numbers $x$ and $y$, $z=x+iy$ is a complex number.
Basic relation: $$i^2=-1,\qquad i\text{ is the imaginary unit}.$$
Example: $\sqrt{-4}=2\sqrt{-1}=2i$

0.4.2 Physical Application

0.4.3 Common inequalities

Basic Inequalities

Multiplication and square root principles:
$$(\text{ Multiplication rule })a>b>0\Rightarrow a^{n}>b^{n}(n\in N,\text{ and }n>1) ;$$
$$\\(\text{ Opening the rules })a>b>0\Rightarrow\sqrt[n]{a}>\sqrt[n]{b}(n\in N,\text{ and }n>1)$$
Basic inequalities:
$$a^2+b^2\geq2ab,\qquad\text{with equality iff }a=b.$$
$$\frac{a+b+c}3\geq\sqrt[3]{abc} (a,b,c\in R^+)$$

Common Inequalities in Advanced Mathematics

Absolute value inequality:
$$-|x|\leqslant x\leqslant|x|$$
$$|a|-|b|\leq|a\pm b|\leq|a|+|b|$$
$$|\sin x|\leqslant1$$
Common Other Inequalities:
$$\begin{aligned}&(1)\ |\sin x|\leq |x|,\quad \forall x\in\mathbb{R},\ \text{with equality at }x=0;\\&(2)\ |x|\leq|\tan x|,\quad \forall x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),\ \text{with equality at }x=0;\\&(3)\ \frac{x}{1+x}\leq\ln(1+x)\leq x,\quad \forall x\geq0.\end{aligned}$$
$$\ln n<\sqrt n$$
Integral inequality:
$$f(x){\leqslant}g(x)\text{ , then }\int_a^bf(x)dx{\leqslant}\int_a^bg(x)dx .$$
$$\left|\int_a^bf(x)dx\right|\leqslant\int_a^b\left|f(x)\right|dx(a<b)$$
$\text{ Let }M\text{ and }m\text{ They are functions }f(x)\text{ In the interval }[a,b]\text{ The maximum and minimum values on the \_ }$
$$\\m(b-a)\leqslant\int_a^bf(x)dx\leqslant M(b-a)(a<b)\text{ (Integral Valuation Theorem) }$$