數學公式大全DSE
數學formula
在這篇文章將會向您展示全港最齊，囊括所有數學核心部分要用的公式。
如果您想只在一篇文章中得到：
- 每一個數學Topic要識的Formula
- 每一條公式的元素
小編建議您看完這篇文章。
我們開始吧！
**注意：這篇文章非常長！請善用目錄
DSE 數學公式 中英對照
恆等式公式
$$\begin{aligned}
&a^{2}-b^{2}&& \equiv(a+b)(a-b) \\&(a+b)^{2}&& \equiv a^{2}+2ab+b^{2} \\&(a-b)^2&& \equiv a^{2}-2ab+b^{2} \\&a^{3}+b^{3}&& \equiv(a+b)(a^{2}-ab+b^{2}) \\&a^{3}-b^{3}&& \equiv(a-b)(a^{2}+ab+b^{2})
\end{aligned}$$
誤差公式
$$
\text { 最大絕對誤差 }=\frac{\text { 最細可量度的單位 }}{2}
$$
絕對誤差= |真確值 – 量度值|
$$
\text { 相對誤差 }=\frac{\text { 最大絕對誤差 }}{\text { 量度值 }}=\frac{\text { 绝對誤差 }}{\text { 真確值 }}
$$
$$
\text { 百份比誤差= 相對誤差 } \times 100 \%
$$
最大值及最少值
- 最大值 = 量度值 + 最大絕對誤差
- 最少值 = 量度值 – 最大絕對誤差
百分比公式
若 x 增加 r%，
$$\text{新值 = x}(1+\mathrm{r}\%)=\mathrm{x}(1+\frac{\mathrm{r}}{100})$$
若x減少r%
$$\text{新值 = x}\mathbf{(1-r\%)=x(1-\frac r{100})}$$
$$\text{百分比改變 =}\mathbf{(\frac {新數值-舊數值}{|舊數值|})}\times 100 \%$$
盈利虧蝕百分率
$$
\text { 盈利或虧蝕百分比 }=\frac{\text { 盈利或虧蝕 }}{\text { 成本 }} \times 100 \%
$$
標價公式
標價及售價
$$
\text { 售價 }=\text { 標價 } \times(1-\text { 折扣\%) }
$$
單利息和複利息
設 P 為本金、R%為利率、
n 為時期、A 為本利和
單利息
$$\text{單利息 = P R% n= }\frac{\mathrm{PRn}}{100}$$
$$\begin{aligned}\mathsf{A}&=\mathsf{P}+\mathsf{I}\\\\&=\mathsf{P}+\frac{\mathsf{PRn}}{100}\\\\&=\mathsf{P}(1+\frac{\mathsf{Rn}}{100})\end{aligned}$$
複利息
$$\mathrm{A=P(1+R\%)^n}$$
$$\text{複利息 }\mathrm{I=A-P}$$
$$=\text{P}(1+\text{R}\%)^n-\text{P}$$
比率與比例公式
$$
\text{平均速率}=\frac{\text{總距離}}{\text{總時間}}
$$
比例尺
\begin{aligned}\text{距離比例}&=1{:}1000\\\\\text{面積比例}&=(1{:}1000)^2\end{aligned}
比率與比例公式
$$
\text{平均速率}=\frac{\text{總距離}}{\text{總時間}}
$$
比例尺
\begin{aligned}\text{距離比例}&=1{:}1000\\\\\text{面積比例}&=(1{:}1000)^2\end{aligned}
指數定律公式
$$\text{已知: }a,b\neq0,m,n\in\mathbb{Q}$$
\begin{aligned}
&a^{n}\times a^{m}=a^{n+m} \\
&{\frac{a^{n}}{a^{m}}}=a^{n-m} \\
&(a^{n})^{m}=a^{n\times m} \\
&a^{-1}=\frac{1}{a} \\
&a^{-n}=\frac{1}{a^{n}}
\end{aligned}
$$\begin{aligned}
&(ab)^{n}=a^{n}\times b^{n} \\
&\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}} \\
&a^{0}=1 \\
&a^{\frac{1}{n}}=\sqrt[n]{a}(\text{n 為正整數})
\end{aligned}$$
坐標幾何公式
$$\begin{aligned}
&\text{兩點距離} =\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}} \\
&\text{斜率}(m)=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\tan\theta \\
&\text{中點} =\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)
\end{aligned}$$
$$\begin{array}{l}{\text{若}L_{1}\perp L_{2},m_{1}\times m_{2}=-1}\\\\{\text{若}L_{1}\parallel L_{2},m_{1}=m_{2}}\\\end{array}$$
$$\begin{array}{l}{\mathrm{Given}:AC{:}CB=m{:}n,A(x_{1},y_{1}),B(x_{2},y_{2})}\\{C=\left(\frac{nx_{1}+mx_{2}}{m+n},\frac{sy_{1}+ry_{2}}{m+n}\right)}\\\end{array}$$
求積法公式
三角形面積公式
$$\text{面積}=\frac12\times\text{底}\times\text{高}\\\text{面積}=\frac12\text{ab}\sin\text{C}$$
$$\text{面積}=\sqrt{s(s-a)(s-b)(s-c)}$$
$$s=\frac{a+b+c}{2}$$
梯形面積公式
$$\text{面積}=\frac{1}{2}\Big(\text{上底}+\text{下底}\Big)\times\text{高}$$
圓形公式
$$\text{面積}=\pi r^{2}$$
$$\text{周界}=2\pi\mathrm{r}$$
扇形弧長 面積公式DSE
$$\begin{aligned}&l\text{ 的弧長 }=2\pi r\times\frac\theta{360°}\\\\&\text{ 扇形 A 的面積 }=\pi r^2\times\frac\theta{360°}\end{aligned}$$
錐體公式
$$\text{體積}=\frac{1}{3}\times\text{底面積}\times\text{高}$$
圓錐體公式
圓錐體積 & 總表面面積公式
\begin{aligned}&\text{體積}=\frac13\pi\mathrm{r}^2\mathrm{h}\\\\&\text{總表面面積}=\pi\mathrm{r}^2+\pi\mathrm{r}\ell\\\\&\text{曲面面積}=\pi\mathrm{r}\ell\end{aligned}
柱體公式
$$\text{體積}=\text{底面積}\times\text{高}$$
圓柱體公式
$$\begin{aligned}&\text{體積}=\pi\mathrm{r}^2\mathrm{h}\\\\&\text{總表面面積}=2\pi\mathrm{r}^2+2\pi\mathrm{r}\mathrm{h}\\\\&\text{曲面面積}=2\pi\mathrm{r}\mathrm{h}\end{aligned}$$
球體公式
$$\begin{aligned}&\text{體積}=\frac43\times\pi\times\mathrm{r}^3\\\\&\text{總表面面積}=4\pi\mathrm{r}^2\end{aligned}$$
三角恆等式公式
- $$\sin^2\theta+\cos^2\theta=1$$
- $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
- $$\sin(90^{\circ}-\theta)=\cos\theta$$
- $$\cos(90^{\circ}-\theta)=\sin\theta$$
- $$\tan(90^{\circ}-\theta)={\frac{1}{\tan\theta}}$$
- $$\sin(90^{\circ}+\theta)=\cos\theta$$
- $$\cos(90^{\circ}+\theta)=-\sin\theta$$
- $$\tan(90^{\circ}+\theta)=-\frac{1}{\tan\theta}$$
- $$\sin(180^{\circ}-\theta)=\sin\theta$$
- $$\cos(180^{\circ}-\theta)=-\cos\theta$$
- $$\tan(180^{\circ}-\theta)=-\tan\theta$$
- $$\sin(180^{\circ}+\theta)=-\sin\theta$$
- $$\cos(180^{\circ}+\theta)=-\cos\theta$$
- $$\tan(180^{\circ}+\theta)=\tan\theta$$
- $$\sin(270^{\circ}-\theta)=-\cos\theta$$
- $$\cos(270^{\circ}-\theta)=-\sin\theta$$
- $$\tan(270^{\circ}-\theta)={\frac{1}{\tan\theta}}$$
- $$\sin(270^{\circ}+\theta)=-\cos\theta$$
- $$\cos(270^{\circ}+\theta)=\sin\theta$$
- $$\tan(270^{\circ}+\theta)=-\frac{1}{\tan\theta}$$
- $$\sin(360^{\circ}-\theta)=-\sin\theta$$
- $$\cos(360^{\circ}-\theta)=\cos\theta$$
- $$\tan(360^{\circ}-\theta)=-\tan\theta$$
- $$\sin(360^{\circ}+\theta)=\sin\theta$$
- $$\cos(360^{\circ}+\theta)=\cos\theta$$
- $$\tan(360^{\circ}+\theta)=\tan\theta$$
三角不等式公式
$$\begin{array}{cc}{1.}&{a+b>c}\\{2.}&{a+c>b}\\{3.}&{b+c>a}\\\end{array}$$
數系公式
根
$$\begin{aligned}
&\text{對任何兩個正數a和b,} \\
&\mathrm{(a)~\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}} \\
&\mathrm{(b)~\sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}}
\end{aligned}$$
複數
$$\begin{aligned}
&若i=\sqrt{-1},a和b 是非零實數,則 \\
&\mathrm{(a)~a\cdot i=i\cdot a=ai} \\
&\mathrm{(b)~ai+bi=(a+b)i} \\
&\text{(c) ai-bi=(a-b)i} \\
&\mathbf{(d)}\text{ ai}\cdot\mathrm{bi}=-\mathrm{ab} \\
&\textbf{(e) }\frac{\mathrm{ai}}{\mathrm{bi}}=\frac{\mathrm{a}}{\mathrm{b}}
\end{aligned}$$
一元二次方程方程式
二次方程求根公式
$$\begin{aligned}\text{若}&ax^2+bx+c=0(a\neq0)\text{,則}\\&x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{aligned}$$
根的性質
$$\begin{aligned}
&\Delta=b^2-4ac 是一元二次方程 \\
&\begin{aligned}ax^2+bx+c=0(a\neq0)\end{aligned}\text{的判別式} \\
& (a) 若\Delta>0,方程有兩個不同實根。 \\
&(b) 若\Delta=0,方程有二重實根。 \\
&(c)若\Delta<0 ,方程沒有實根 。
\end{aligned}$$
兩根之和 兩根之積
$$\begin{aligned}&\alpha+\beta=-\frac{\mathrm{b}}{\mathrm{a}}\quad;\quad\alpha\beta=\frac{\mathrm{c}}{\mathrm{a}}\\&\text{方程式}\\&\mathbf{x}^2-(\alpha+\beta)\mathbf{x}+\alpha\beta=0\end{aligned}$$
二次函數頂點
$$\begin{aligned}\text{二次方程:}y&=ax^2+bx+c\\\\&\textit{頂點}=\left(-\frac{b}{2a},-\frac{b^2-4ac}{4a}\right)\end{aligned}$$
多項式
餘式定理
$$\text{若}\mathrm{f(x)}\text{除以mx}-\mathrm{n}\text{ , 餘式是f}\mathrm{\left(\frac nm\right)}\text{。}$$
因式定理
$$\begin{aligned}
&\text{(a)若f}\left(\frac n{\mathfrak{m}}\right)=0\text{ ,則m}\mathfrak{x}-\mathfrak{n}\text{是f}(\mathfrak{x})\text{的因式 }。 \\
&\text{(b) 若mx-n是f(x)的因式,則} \\
&\mathbf{f}\left(\frac{n}{\mathfrak{m}}\right)=\mathbf{0}\circ
\end{aligned}$$
Log公式大全
Log Formula DSE
$$\begin{aligned}
&當M和N是正數,a>0,a\neq1和k是 \\
&\text{有理數,則} \\
&(\mathbf{a})\log_{a}a^{k}=k \\
&(\mathbf{b})\log_{a}a=1 \\
&(\mathbf{c})\log_{a}1=0 \\
& (\mathbf{d})\log_{a}MN=\log_{a}M+\log_{a}N \\
&\mathbf{(e)}\log_{a}{\frac{M}{N}}=\log_{a}M-\log_{a}N \\
&(\mathbf{f})\log_{a}M^{k}=k\log_{a}M \\
&(\mathbf{g})\log_{a}M=\frac{\log_{b}M}{\log_{b}a}(b>0\mathrm{~and~}b\neq1)
\end{aligned}$$
地震强度公式
地震的強度可以用芮氏地震等級(Richter magnitude scale)來衡量。芮氏地震等級(M)的計算公式為:
M = log(A) – log(A0)
其中:
- A 是地震波在某一距離處的最大振幅
- A0 是基準振幅
聲音强度公式
分貝(decibel, dB)是測量聲強或功率的對數單位。分貝公式為:
dB = 10 log(P/P0)
其中:
- P 是待測聲音的功率
- P0 是參考聲功率(通常取 10^-12 瓦)
這個公式描述了聲音功率與分貝值之間的換算關係。分貝值越高,表示聲音越大。
變分
正變
$$\begin{aligned}&\text{若у隨x而正變，則 }\mathbf{y=kx}\text{，其中 k 是}\\&\text{非零常數}\end{aligned}$$
反變
$$\begin{aligned}&\text{若у隨x而反變，則}\mathbf{y=\frac kx}\text{，其中 k 是}\\&\text{非零常數}\end{aligned}$$
聯變
$$\begin{aligned}
&\text{(a)若z 隨x和y而聯變,則z=kxy,其} \\
&\text{中k是非零常數。} \\
&\text{(b) 若z 隨x 而正變和隨y而反變;則} \\
&\mathrm{z}=\frac{kx}{\mathrm{y}}\text{,其中 k }\text{ 是非零常數 }\circ
\end{aligned}$$
部分變公式
$$\begin{aligned}
&\text{(a)若z 部分為常數;部分隨x而正變,} \\
&則z=k_1+k_2x;其中k_1和k_2\text{是非零常} \\
&\text{數。} \\
&\text{(b) 若z 部分隨 x而正變和部分隨 y而} \\
&\text{反變,則}\mathrm{z=k_1x+\frac{k_2}{y}}\text{,其中}\mathrm{k_1}\text{和}\mathrm{k_2}\text{是} \\
&\text{非零常數 。}
\end{aligned}$$
不等式公式
\begin{aligned}
\text{(a) 若a>b} \text{和 b>c, 則 a>c 。}\\
\text{(b) 若a>b, 則 a+c>b+c 。} \\
\text{(c) 若a>b} \text{,和} \\
\text{(i)c>0} \text{,則ac>bc;} \\
\text{(i)c<0} \text{,則ac<bc°} \\
\text{(d) 若 a>b>0,則}\frac1{\mathrm{a}}<\frac1{\mathrm{b}}\circ \\
\mathsf{(e)}\textit{若}\mathfrak{a}\neq0\text{,則}\mathfrak{a}^2>0\circ
\end{aligned}
三角形公式表
sin cos tan Formula DSE
$$\begin{aligned}
\sin\theta={\frac{BC}{AC}} \\
\cos\theta={\frac{AB}{AC}} \\
\tan\theta={\frac{BC}{AB}}
\end{aligned}$$
畢氏定理
$$\begin{aligned}&\text{若}\angle\mathrm{ACB}=90^\circ\quad\text{,}\\&\text{則}\mathtt{c}^2=\mathtt{a}^2+\mathtt{b}^2\quad\text{(畢氏定理)}\end{aligned}$$
畢氏定理逆定理
$$\begin{aligned}&\text{若}\mathbf{c}^2=\mathbf{a}^2+\mathbf{b}^2\text{,}\\&\text{則}\angle\mathrm{ACB}=90°\textbf{(畢氏定理逆定理})\end{aligned}$$
正弦公式
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
餘弦公式
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
續函數及其圖像
| 代數上的函數變換 | 幾何上的函數變換 |
|---|---|
| f(x) + k | 向上平移 k 單位 |
| f(x) – k | 向下平移 k 單位 |
| f(x + k) | 向左平移 k 單位 |
| f(x – k) | 向右平移 k 單位 |
| k * f(x) | 沿 y-軸放大至原來的 k 倍 |
| (1/k) * f(x) | 沿 y-軸縮小至原來的 k 倍 |
| f(-x) | 沿 y-軸反射 |
| -f(x) | 沿 x-軸反射 |
| f(kx) | 沿 x-軸縮小至原來的 1 倍 k |
| f(kx) | 沿 x-軸放大至原來的 1 倍 k |
圓的方程DSE
$$\begin{aligned}
&\text{圓的方程} \\
&\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2} \\
&圓心為 (h,k) \\
&\text{半徑 =r} \\
&\text{圓的方程} \\
&\text{一般式:}x^{2}+y^{2}+Dx+Ey+F=0 \\
&\text{圓心為 }(-\frac{D}{2},-\frac{E}{2}) \\
&\text{半徑 }=\sqrt{(\frac D2)^{2}+(\frac E2)^{2}-F}
\end{aligned}$$
排列與組合
階乘
$$\begin{aligned}&\mathrm{n!=~n(n-1)(n-2)~\cdot~\cdot~\cdot~\cdot~3\cdot~2\cdot~1}\\&\text{,其中 n 是一個正整數}\end{aligned}$$
nPr
$$\begin{aligned}
&\text{從n個相異物中,任意選取r個} \\
&( 0<r\leq n ,不許重複),然後按序作 \\
&\text{直線排列(其排列總數記為}P_r^n\quad, \\
&P_{r}^{n}=\frac{n!}{(n-r)!}
\end{aligned}$$
nCr
$$\begin{aligned}
&\text{從n個相異物中,不按序的任意選取r} \\
&個(0<r\leq n ,不許重複),其組合數 \\
&\text{為}C_r^n\quad, \\
&C_{r}^{n}=\frac{P_{r}^{n}}{r!}=\frac{n!}{r!(n-r)!}
\end{aligned}$$
概率
$$\begin{aligned}
&\text{(a)}&& P(E)=\frac{\text{符合事件 }E\text{ 的結果數目}}{\text{可能結果的總數}} , \\
&&&\text{其中所有可能結果發生的機會均等。} \\
&\mathbf{(b)}&& P(\text{必然事件 })=1 \\
&\mathbf{(c)}&& P(\text{不可能事件})=0 \\
&(\mathbf{d})&& 0\leq P(E)\leq1
\end{aligned}$$
期望值
$$\begin{aligned}
&\text{假設某事件有n個結果,而每個結果} \\
&發生的概率分別為p_1,p_2,…,p_n°若每 \\
&\text{個結果發生後可取的值分別為} \\
&x_1,x_2,…,x_n , 則該事件的期望值 \\
&\mathbf{=}\mathbf{x}_1\mathbf{p}_1+\mathbf{x}_2\mathbf{p}_2+\cdots+\mathbf{x}_n\mathbf{p}_n
\end{aligned}$$
加法定律
$$\begin{aligned}
&\text{若A和B不可能同時發生,} \\
&\text{則稱為互斥事件 。} \\
&\text{(a) 若A和B是互斥事件,則} \\
&\mathsf{P(A~或~B)=P(A)+P(B)} \\
&\text{(b)若A和B不是互斥事件,則} \\
&\text{P(A 或 B)=P(A)+P(B)-P(A 和 B)} \\
&(c) 對任何事件 A ; P(A)+P(A^{\prime})=1 , \\
&\text{其中 A’是A的互補事件 。}
\end{aligned}$$
乘法定律
$$\begin{aligned}&\text{若A和B 是獨立事件,}\\&\text{則 P(A 和 B)}=\mathsf{P(A)\times P(B)}\end{aligned}$$
條件概率
$$\begin{aligned}
&\text{已知A發生,B 發生的機會率} \\
&=\mathcal{P}(\mathcal{B}|\mathcal{A}) \\
&=\frac{P(A\text{ 及 }B)}{P(A)},\text{其中}P(A)\neq0
\end{aligned}$$
AS GS
等差數列公式 DSE
$$\begin{aligned}
&\mathbf{l.}\textbf{ 等差數列}{ : }\mathbf{a,a+d,a+2d\ldots} \\
&\text{(a) T(n) = a+(n-1)d} \\
&\textbf{(b) T(n) = }\frac12\text{[T(n-1)+T(n+1)]} \\
&(c) 若 T(1),T(2)…是等差數列;則 \\
&\mathrm{kT(1)+c,~kT(2)~+c}\text{都是等差} \\
&\text{數列} \\
&\text{(d) 已知 S(n)=T(1)+T(2)+…+T(n)} \\
&\mathbf{S(n)=\frac{n}{2}[a+T(n)]\\=\frac{n}{2}[2a+(n-1)d]}
\end{aligned}$$
等比數列公式 DSE
$$\begin{aligned}
&\textbf{II. 等比數列: }\mathbf{a,ar^2,ar^3,ar^4,…} \\
&\mathrm{(a)~T(n)=ar^{n-1}} \\
&\mathrm{(b)~T(n)=~\sqrt{T(n-1)\times T(n+1)}} \\
&(c) 若 T(1),T(2),T(3),…是等比數列;則 \\
&\mathrm{kT}(1),\mathrm{kT}(2),\mathrm{kT}(3),…\text{都是等比數列 }\\
&\mathrm{(d)~}\text{ 已知 S(n)=T(1)+T(2)+…+T(n)} \\
&S(n)={\frac{a(r^{n}-1)}{r-1}}{\text{或}}S(n)={\frac{a(1-r^{n})}{1-r}} \\
&,\text{其中}r\neq0\text{及}r\neq1\circ \\
&\text{(e)} \\
&\text{已知 S}(\infty)=T(1)+\cdots+T(n)+\cdots\\\
&S(\infty)=\frac{a}{1-r},-1<r<1\text{及}r\neq0
\end{aligned}$$
統計學公式
平均值
$$\overline{x} = $$
$$\frac{x_{1}, x_{2}, x_{3}, \ldots, x_{n}}{n}$$
中位數
如果數值的總個數是奇數
$$\left(\frac{n+1}{2}\right)$$
如果數值的總個數是偶數
$$\left(\frac{n}{2}\right)$$ 和 $$\left(\frac{n$}{2}+1\right)$$ 的兩個數值的平均值
分佈域
分佈域 = 最大值 – 最小值
四分位間距
四分位間距 = 上四分位數 – 下四分位數
標準差
$$\sigma=\sqrt{\frac{1}{n}{\sum_{k=1}^n(x_i-\bar{x})^2}}$$
方差
$$\sigma^2=\frac{1}{n}{\sum_{k=1}^n(x_i-\bar{x})^2}$$
標準分
$$\text{標準分} =\frac{x_{n}-\bar{\mathrm{x}}}{\sigma}$$
正態分佈
在正態分佈中，
1sigma 百分比
(i) 大約有 68%數據位於
$$\bar{\mathbf{x}}-\sigma$$和$$\bar{\mathbf{x}}+\sigma$$之間
2sigma 百分比
(ii) 大約有 95%數據位於$$\bar{\mathbf{x}}-2\sigma$$和$$\bar{\mathbf{x}}+2\sigma$$之間
3sigma 百分比
(iii) 大約有 99.7%數據位於$$\bar{\mathrm{x}}-3\sigma$$和
$$\bar{\mathrm{x}}+3\sigma$$之間
數據改變
| 全部+k | 全部 X k |
平均數、中位數、眾數 | 全部+k | 全部 X k |
標準差、分佈域、四分位數間距 | 不變 | 全部 X k |
Common Identity
$$\begin{aligned}
&a^{2}-b^{2}&& \equiv(a+b)(a-b) \\&(a+b)^{2}&& \equiv a^{2}+2ab+b^{2} \\&(a-b)^2&& \equiv a^{2}-2ab+b^{2} \\&a^{3}+b^{3}&& \equiv(a+b)(a^{2}-ab+b^{2}) \\&a^{3}-b^{3}&& \equiv(a-b)(a^{2}+ab+b^{2})
\end{aligned}$$
Estimation and Error Formula
$$
\text { Max. Abs. Error }=\frac{\text
{ Smallest Unit}}{2}
$$
Abs. Error = |True Value – Measured Value|
$$
\text { Rel. Error}=\frac{\text { Max. Abs. Error }}{\text { Measured Value }}\\=\frac{\text { Abs. Error }}{\text { True Value }}
$$
$$
\text { % Error = Rel. Error } \times 100 \%
$$
Max. Value & Min. Value
- Max. Value = Measured Value + Max. Abs. Value
- Min. Value = Measured Value – Max. Abs. Value
Percentage Formula
If x increases by r%，
$$\text{New Value = x}(1+\mathrm{r}\%)=\mathrm{x}(1+\frac{\mathrm{r}}{100})$$
If x decreases by r%
$$\text{New Value = x}\mathbf{(1-r\%)=x(1-\frac r{100})}$$
$$\text{% Change=}\mathbf{(\frac {New Value-Original Value}{Original Value})}\times 100 \%$$
Profit percentage
$$
\text {% Profit / Loss }=\frac{\text { Profit/Loss }}{\text {Cost}} \times 100 \%
$$
Marked Price & Selling Price
$$
\text { Selling Price }=\text { Marked Price } \times(1-\text { discount\%) }
$$
Simple Interest and Compound Interest
P=Principal、R%=Rate、
n=Time、A=Amount
Simple Interest
$$\text{Interest = P R% n= }\frac{\mathrm{PRn}}{100}$$
$$\begin{aligned}\mathsf{A}&=\mathsf{P}+\mathsf{I}\\\\&=\mathsf{P}+\frac{\mathsf{PRn}}{100}\\\\&=\mathsf{P}(1+\frac{\mathsf{Rn}}{100})\end{aligned}$$
Compound Interest
$$\mathrm{A=P(1+R\%)^n}$$
$$\text{Compound Interest}\mathrm{I=A-P}$$
$$=\text{P}(1+\text{R}\%)^n-\text{P}$$
Rate and Ratio Formula
$$
\text{Avg. speed}=\frac{\text{Total Distance}}{\text{Total time}}
$$
Map Scale
\begin{aligned}\text{Distance Ratio}&=1{:}1000\\\\\text{Area Ratio}&=(1{:}1000)^2\end{aligned}
Law of Indices Formula
$$\text{Given: }a,b\neq0,m,n\in\mathbb{Q}$$
\begin{aligned}
&a^{n}\times a^{m}=a^{n+m} \\
&{\frac{a^{n}}{a^{m}}}=a^{n-m} \\
&(a^{n})^{m}=a^{n\times m} \\
&a^{-1}=\frac{1}{a} \\
&a^{-n}=\frac{1}{a^{n}}
\end{aligned}
$$\begin{aligned}
&(ab)^{n}=a^{n}\times b^{n} \\
&\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}} \\
&a^{0}=1 \\
&a^{\frac{1}{n}}=\sqrt[n]{a}(\text{n 為+ve Integer})
\end{aligned}$$
Coordinate Geometry Formula
$$\begin{aligned}
&\text{Distance} =\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}} \\
&\text{Slope}(m)=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\tan\theta \\
&\text{Mid-point} =\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)
\end{aligned}$$
$$\begin{array}{l}{\text{If}L_{1}\perp L_{2},m_{1}\times m_{2}=-1}\\\\{\text{If}L_{1}\parallel L_{2},m_{1}=m_{2}}\\\end{array}$$
$$\begin{array}{l}{\mathrm{Given}:AC{:}CB=m{:}n,A(x_{1},y_{1}),B(x_{2},y_{2})}\\{C=\left(\frac{nx_{1}+mx_{2}}{m+n},\frac{sy_{1}+ry_{2}}{m+n}\right)}\\\end{array}$$
Mensuration Formula DSE
Triangle
$$\text{Area}=\frac12\times\text{Base}\times\text{Height}\\\text{Area}=\frac12\text{ab}\sin\text{C}$$
$$\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$$
$$s=\frac{a+b+c}{2}$$
Trapezium
$$\text{Area}=\frac{1}{2}\Big(\text{Upper base}+\text{Lower base}\Big)\times\text{Height}$$
Circle
$$\text{Area}=\pi r^{2}$$
$$\text{Perimeter}=2\pi\mathrm{r}$$
Sector
$$\begin{aligned}&l\text{ arc }=2\pi r\times\frac\theta{360°}\\\\&\text{ Area of sector }=\pi r^2\times\frac\theta{360°}\end{aligned}$$
Pyramid
$$\text{Volume}=\frac{1}{3}\times\text{Base area}\times\text{Height}$$
Cone
\begin{aligned}&\text{Volume}=\frac13\pi\mathrm{r}^2\mathrm{h}\\\\&\text{Total Surface area}=\pi\mathrm{r}^2+\pi\mathrm{r}\ell\\\\&\text{Curve surface area}=\pi\mathrm{r}\ell\end{aligned}
Prism
$$\text{Volume}=\text{Base area}\times\text{Height}$$
Cylinder
$$\begin{aligned}&\text{體積}=\pi\mathrm{r}^2\mathrm{h}\\\\&\text{Total surface area}=2\pi\mathrm{r}^2+2\pi\mathrm{r}\mathrm{h}\\\\&\text{curve surface area}=2\pi\mathrm{r}\mathrm{h}\end{aligned}$$
Sphere
$$\begin{aligned}&\text{Volume}=\frac43\times\pi\times\mathrm{r}^3\\\\&\text{Total surface area}=4\pi\mathrm{r}^2\end{aligned}$$
Trigonometry Identities Formula
- $$\sin^2\theta+\cos^2\theta=1$$
- $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
- $$\sin(90^{\circ}-\theta)=\cos\theta$$
- $$\cos(90^{\circ}-\theta)=\sin\theta$$
- $$\tan(90^{\circ}-\theta)={\frac{1}{\tan\theta}}$$
- $$\sin(90^{\circ}+\theta)=\cos\theta$$
- $$\cos(90^{\circ}+\theta)=-\sin\theta$$
- $$\tan(90^{\circ}+\theta)=-\frac{1}{\tan\theta}$$
- $$\sin(180^{\circ}-\theta)=\sin\theta$$
- $$\cos(180^{\circ}-\theta)=-\cos\theta$$
- $$\tan(180^{\circ}-\theta)=-\tan\theta$$
- $$\sin(180^{\circ}+\theta)=-\sin\theta$$
- $$\cos(180^{\circ}+\theta)=-\cos\theta$$
- $$\tan(180^{\circ}+\theta)=\tan\theta$$
- $$\sin(270^{\circ}-\theta)=-\cos\theta$$
- $$\cos(270^{\circ}-\theta)=-\sin\theta$$
- $$\tan(270^{\circ}-\theta)={\frac{1}{\tan\theta}}$$
- $$\sin(270^{\circ}+\theta)=-\cos\theta$$
- $$\cos(270^{\circ}+\theta)=\sin\theta$$
- $$\tan(270^{\circ}+\theta)=-\frac{1}{\tan\theta}$$
- $$\sin(360^{\circ}-\theta)=-\sin\theta$$
- $$\cos(360^{\circ}-\theta)=\cos\theta$$
- $$\tan(360^{\circ}-\theta)=-\tan\theta$$
- $$\sin(360^{\circ}+\theta)=\sin\theta$$
- $$\cos(360^{\circ}+\theta)=\cos\theta$$
- $$\tan(360^{\circ}+\theta)=\tan\theta$$
Triangle Inequality
$$\begin{array}{cc}{1.}&{a+b>c}\\{2.}&{a+c>b}\\{3.}&{b+c>a}\\\end{array}$$
Trigonometry
$$\begin{aligned}
\sin\theta={\frac{BC}{AC}} \\
\cos\theta={\frac{AB}{AC}} \\
\tan\theta={\frac{BC}{AB}}
\end{aligned}$$
Pythagoras theorem
$$\begin{aligned}&\text{If}\angle\mathrm{ACB}=90^\circ\quad\text{,}\\&\text{則}\mathtt{c}^2=\mathtt{a}^2+\mathtt{b}^2\quad\text{Pythagoras theorem)}\end{aligned}$$
Converse of Pythagoras theorem
$$\begin{aligned}&\text{If}\mathbf{c}^2=\mathbf{a}^2+\mathbf{b}^2\text{,}\\&\text{則}\angle\mathrm{ACB}=90°\textbf{(Converse of Pythagoras theorem})\end{aligned}$$
Sine formula
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
Cosine formula
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
Number system Formula
Surd
$$\begin{aligned}
&\text{For any two +ve numbers a和b,} \\
&\mathrm{(a)~\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}} \\
&\mathrm{(b)~\sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}}
\end{aligned}$$
Complex Number
$$\begin{aligned}
&If i=\sqrt{-1}, a & b are non zero real number, then \\
&\mathrm{(a)~a\cdot i=i\cdot a=ai} \\
&\mathrm{(b)~ai+bi=(a+b)i} \\
&\text{(c) ai-bi=(a-b)i} \\
&\mathbf{(d)}\text{ ai}\cdot\mathrm{bi}=-\mathrm{ab} \\
&\textbf{(e) }\frac{\mathrm{ai}}{\mathrm{bi}}=\frac{\mathrm{a}}{\mathrm{b}}
\end{aligned}$$
Quadratic Equation Formula
Quadratic Formula
$$\begin{aligned}\text{若}&ax^2+bx+c=0(a\neq0)\text{,則}\\&x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\end{aligned}$$
Discriminant
$$\begin{aligned}
&\Delta=b^2-4ac is quadratic eqution\\
&\begin{aligned}ax^2+bx+c=0(a\neq0)\end{aligned}\text{Discriminant} \\
& (a) If\Delta>0, two real roots \\
&(b) If\Delta=0, repeated roots \\
&(c) If\Delta<0, no real roots
\end{aligned}$$
Sum of roots Product of roots
$$\begin{aligned}&\alpha+\beta=-\frac{\mathrm{b}}{\mathrm{a}}\quad;\quad\alpha\beta=\frac{\mathrm{c}}{\mathrm{a}}\\&\text{Equation}\\&\mathbf{x}^2-(\alpha+\beta)\mathbf{x}+\alpha\beta=0\end{aligned}$$
Vertex
$$\begin{aligned}\text{Quadratic Function:}y&=ax^2+bx+c\\\\&\textit{Vertex}=\left(-\frac{b}{2a},-\frac{b^2-4ac}{4a}\right)\end{aligned}$$
Polynomial
Remainder theorem
$$\text{If}\mathrm{f(x)}\text{is divided mx}-\mathrm{n}\text{ , Remainder is f}\mathrm{\left(\frac nm\right)}\text{。}$$
Factor theorem
$$\begin{aligned}
&\text{(a)If f}\left(\frac n{\mathfrak{m}}\right)=0\text{ ,then m}\mathfrak{x}-\mathfrak{n}\text{is f}(\mathfrak{x})\text{factor }。 \\
&\text{(b) If mx-n is factor of f(x), then} \\
&\mathbf{f}\left(\frac{n}{\mathfrak{m}}\right)=\mathbf{0}\circ
\end{aligned}$$
Log Formula DSE
Logarithmic Function Formula
$$\begin{aligned}
&When M & N are real number, a>0,a\neq1 & k is \\
&\text{real number, then} \\
&(\mathbf{a})\log_{a}a^{k}=k \\
&(\mathbf{b})\log_{a}a=1 \\
&(\mathbf{c})\log_{a}1=0 \\
& (\mathbf{d})\log_{a}MN=\log_{a}M+\log_{a}N \\
&\mathbf{(e)}\log_{a}{\frac{M}{N}}=\log_{a}M-\log_{a}N \\
&(\mathbf{f})\log_{a}M^{k}=k\log_{a}M \\
&(\mathbf{g})\log_{a}M=\frac{\log_{b}M}{\log_{b}a}(b>0\mathrm{~and~}b\neq1)
\end{aligned}$$
Log Formula DSE
Logarithmic Function Formula
$$\begin{aligned}
&When M & N are real number, a>0,a\neq1 & k is \\
&\text{real number, then} \\
&(\mathbf{a})\log_{a}a^{k}=k \\
&(\mathbf{b})\log_{a}a=1 \\
&(\mathbf{c})\log_{a}1=0 \\
& (\mathbf{d})\log_{a}MN=\log_{a}M+\log_{a}N \\
&\mathbf{(e)}\log_{a}{\frac{M}{N}}=\log_{a}M-\log_{a}N \\
&(\mathbf{f})\log_{a}M^{k}=k\log_{a}M \\
&(\mathbf{g})\log_{a}M=\frac{\log_{b}M}{\log_{b}a}(b>0\mathrm{~and~}b\neq1)
\end{aligned}$$
Richter magnitude scale
The intensity of an earthquake is measured using the Richter magnitude scale. This scale is based on the amplitude of seismic waves, which represents the size of the earthquake. The amplitude of seismic waves is measured using a seismograph.
The formula to calculate the Richter magnitude (M) is:
M = log(A) – log(A0)
Here, A represents the maximum amplitude of seismic waves at a certain distance, and A0 is a reference amplitude.
In this formula, we take the logarithm of the maximum amplitude A and the reference amplitude A0, and then calculate the difference between them. This difference gives us the Richter magnitude M.
Sound Intensity Formula
The decibel (dB) is a logarithmic unit used to measure sound intensity or power. The decibel formula is:
dB = 10 log(P/P0)
Here:
P represents the power of the sound being measured.
P0 is the reference power level (typically set at 10^-12 watts).
This formula describes the conversion relationship between sound power and decibel value. A higher decibel value indicates a louder sound.
By taking the ratio of the power P to the reference power P0, applying a logarithm to the base 10, and multiplying by 10, we obtain the decibel value. This logarithmic scale allows for a more convenient representation of a wide range of sound intensities.
Variation
Directly Proportional
$$\begin{aligned}&\text{If у is proportional to x, then }\mathbf{y=kx}\text{, k is}\\&\text{non-zero constant}\end{aligned}$$
Inversely Proportional
$$\begin{aligned}&\text{If у is inversely proportional to x, then }\mathbf{y=kx}\text{, k is}\\&\text{non-zero constant}\end{aligned}$$
Joint Variation
$$\begin{aligned}
&\text{(a)If z varies with x and y, then z=kxy,} \\
&\text{k is non-zero constant} \\
&\text{(b) If z directly varies with x and } \\
&\mathrm{z}=\frac{kx}{\mathrm{y}}\text{, k }\text{ is non-zero constant }\circ
\end{aligned}$$
Partial Variation
$$\begin{aligned}
&\text{(a)If z partly constant and partly varies with x} \\
&then z=k_1+k_2x; k_1和k_2\text{are non-zero constant} \\
&\text{(b) If z partly varies with x and partly inversely varies with y} \\
&\mathrm{z=k_1x+\frac{k_2}{y}}\mathrm{k_1}\text{&}\mathrm{k_2}\text{are} \\
&\text{non-zero constant}
\end{aligned}$$
Inequality Formula
\begin{aligned}
\text{(a) If a>b} \text{和 b>c, If a>c 。}\\
\text{(b) If a>b, If a+c>b+c 。} \\
\text{(c) If a>b} \text{,and} \\
\text{(i)c>0} \text{,If ac>bc;} \\
\text{(i)c<0} \text{,If ac<bc°} \\
\text{(d) If a>b>0,If }\frac1{\mathrm{a}}<\frac1{\mathrm{b}}\circ \\
\mathsf{(e)}\textit{If }\mathfrak{a}\neq0\text{,If }\mathfrak{a}^2>0\circ
\end{aligned}
Transformation of Function
| Algebraic Change | Geometric Change |
|---|---|
| f(x) + k | 向上平移 k 單位 |
| f(x) – k | 向下平移 k 單位 |
| f(x + k) | 向左平移 k 單位 |
| f(x – k) | 向右平移 k 單位 |
| k * f(x) | 沿 y-軸放大至原來的 k 倍 |
| (1/k) * f(x) | 沿 y-軸縮小至原來的 k 倍 |
| f(-x) | 沿 y-軸反射 |
| -f(x) | 沿 x-軸反射 |
| f(kx) | 沿 x-軸縮小至原來的 1 倍 k |
| f(kx) | 沿 x-軸放大至原來的 1 倍 k |
Equation of Circles
$$\begin{aligned}
&\text{Equation of Circles} \\
&\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2} \\
&Centre為 (h,k) \\
&\text{Radius =r} \\
&\text{Equations} \\
&\text{General Form:}x^{2}+y^{2}+Dx+Ey+F=0 \\
&\text{Centre }(-\frac{D}{2},-\frac{E}{2}) \\
&\text{Radius }=\sqrt{(\frac D2)^{2}+(\frac E2)^{2}-F}
\end{aligned}$$
Combination and Permutation
Factorial
$$\begin{aligned}&\mathrm{n!=~n(n-1)(n-2)~\cdot~\cdot~\cdot~\cdot~3\cdot~2\cdot~1}\\&\text{,n is +ve integer}\end{aligned}$$
nPr
$$\begin{aligned}
&\text{Select r arbitrarily from n different objects} \\
&(0<r\leq n, no repetitions allowed), then do \\ in order
&\text{Linear arrangement (the total number of arrangements is recorded as}P_r^n\quad, \\
&P_{r}^{n}=\frac{n!}{(n-r)!}
\end{aligned}$$
nCr
$$\begin{aligned}
&\text{Select r arbitrarily from n different objects in no order} \\
& (0<r\leq n, no repetitions allowed), the number of combinations \\
&\text{is}C_r^n\quad, \\
&C_{r}^{n}=\frac{P_{r}^{n}}{r!}=\frac{n!}{r!(n-r)!}
\end{aligned}$$
Probability
$$\begin{aligned}
&\text{(a)}&& P(E)=\frac{\text{Number of results matching event }E\text{}}{\text{Total number of possible results}}, \\
&&&\text{All possible outcomes have an equal chance of occurring. } \\
&\mathbf{(b)}&& P(\text{Necessary event})=1 \\
&\mathbf{(c)}&& P(\text{impossible event})=0 \\
&(\mathbf{d})&& 0\leq P(E)\leq1
\end{aligned}$$
Expected Value
$$\begin{aligned}
&\text{Suppose an event has n results, and each result} \\
&The probability of occurrence is p_1,p_2,…,p_n°if every \\
&\text{The possible values after the occurrence of each result are} \\
&x_1,x_2,…,x_n , then the expected value of the event \\
&\mathbf{=}\mathbf{x}_1\mathbf{p}_1+\mathbf{x}_2\mathbf{p}_2+\cdots+\mathbf{x}_n\mathbf{p}_n
\end{aligned}$$
Additive Rule
$$\begin{aligned}
&\text{If A and B cannot happen at the same time,} \\
&\text{is called a mutually exclusive event. } \\
&\text{(a) If A and B are mutually exclusive events, then} \\
&\mathsf{P(A~or~B)=P(A)+P(B)} \\
&\text{(b)If A and B are not mutually exclusive events, then} \\
&\text{P(A or B)=P(A)+P(B)-P(A and B)} \\
&(c) for any event A ; P(A)+P(A^{\prime})=1 , \\
&\text{Where A’ is the complementary event of A. }
\end{aligned}$$
Product Rule
$$\begin{aligned}&\text{If A and B are independent events,}\\&\text{Then P(A and B)}=\mathsf{P(A)\times P(B)}\ end{aligned}$$
AS GS
Arithmetic Sequence
$$\begin{aligned}
&\mathbf{l.}\textbf{ Arithmetic sequence}{ : }\mathbf{a,a+d,a+2d\ldots} \\
&\text{(a) T(n) = a+(n-1)d} \\
&\textbf{(b) T(n) = }\frac12\text{[T(n-1)+T(n+1)]} \\
&(c) If T(1),T(2)… is an arithmetic sequence; then \\
&\mathrm{kT(1)+c,~kT(2)~+c}\text{all arithmetic} \\
&\text{Sequence} \\
&\text{(d) It is known that S(n)=T(1)+T(2)+…+T(n)} \\
&\mathbf{S(n)=\frac{n}{2}[a+T(n)]\\=\frac{n}{2}[2a+(n-1)d]}
\end{aligned}$$
Geometric Sequence
$$\begin{aligned}
&\textbf{II. Geometric sequence: }\mathbf{a,ar^2,ar^3,ar^4,…} \\
&\mathrm{(a)~T(n)=ar^{n-1}} \\
&\mathrm{(b)~T(n)=~\sqrt{T(n-1)\times T(n+1)}} \\
&(c) If T(1),T(2),T(3),… are geometric sequences; then \\
&\mathrm{kT}(1),\mathrm{kT}(2),\mathrm{kT}(3),…\text{They are all geometric sequences}\\
&\mathrm{(d)~}\text{ Known S(n)=T(1)+T(2)+…+T(n)} \\
&S(n)={\frac{a(r^{n}-1)}{r-1}}{\text{or}}S(n)={\frac{a(1-r^{n })}{1-r}} \\
&,\text{where}r\neq0\text{and}r\neq1\circ \\
&\text{(e)} \\
&\text{Known S}(\infty)=T(1)+\cdots+T(n)+\cdots\\\
&S(\infty)=\frac{a}{1-r},-1<r<1\text{and}r\neq0
\end{aligned}$$
Statistics Formula
$$\begin{aligned}
&\text{average}\overline{(\mathrm{x})} =\frac{x_{1}+x_{2}+x_{3}+\cdots+x_{n}}{n} \\
&\text{Median }=\text{th}\left(\frac{n+1}{2}\right)\text{item} \\
&\text{Distribution domain = maximum value-minimum value} \\
&\text{Interquartile range = upper quartile – lower quartile} \\
&\sigma=\sqrt{\frac{(x_{1}-\bar{x})^{2}+\cdots+(x_{n}-\bar{x})^{2}}{n}} \\
&\text{variance}=\frac{(x_{1}-\bar{x})^{2}+\cdots+(x_{n}-\bar{x})^{2}}{n} \ \
&\text{standard score} =\frac{x_{n}-\bar{\mathrm{x}}}{\sigma}
\end{aligned}$$
Normal Distribution
In a normal distribution,
(i) Approximately 68% of the data is located in
Between $$\bar{\mathbf{x}}-\sigma$$ and $$\bar{\mathbf{x}}+\sigma$$
(ii) About 95% of the data lies between $$\bar{\mathbf{x}}-2\sigma$$ and $$\bar{\mathbf{x}}+2\sigma$$
(iii) About 99.7% of the data are located at $$\bar{\mathrm{x}}-3\sigma$$ and
Between $$\bar{\mathrm{x}}+3\sigma$$
Change in data
All+k | All X k | |
mean, median, mode | All+k | All X k |
Standard deviation, range, interquartile range | 不變 | All X k |