x = − b ± b 2 − 4 a c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} a 2 − b 2 = ( a + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=\left(a+b\right)\left(a-b\right)} a 3 ± b 3 = ( a ± b ) ( a 2 ∓ a b + b 2 ) {\displaystyle a^{3}\pm b^{3}=\left(a\pm b\right)\left(a^{2}\mp ab+b^{2}\right)} log a x y = log a x + log a y {\displaystyle \log _{a}xy=\log _{a}x+\log _{a}y} log a x y = log a x − log a y {\displaystyle \log _{a}{\frac {x}{y}}=\log _{a}x-\log _{a}y} log a x m = m log a x {\displaystyle \log _{a}x^{m}=m\log _{a}x} a log a x = x {\displaystyle a^{\log _{a}x}=x} log a x = log b x log b a {\displaystyle \log _{a}x={\frac {\log _{b}x}{\log _{b}a}}} a _ ⋅ b _ = | a _ | | b _ | cos θ {\displaystyle {\underline {a}}\cdot {\underline {b}}=\left|{\underline {a}}\right|\left|{\underline {b}}\right|\cos {\theta }} ( a + b ) n = ∑ r = 0 n n C r a n − r b r {\displaystyle \left(a+b\right)^{n}=\sum _{r=0}^{n}{_{n}}C_{r}a^{n-r}b^{r}} ( 1 + x ) n = 1 + n 1 ! x + n ( n − 1 ) 2 ! x 2 + ⋯ + n ( n − 1 ) … ( n − r + 1 ) r ! x r + … {\displaystyle \left(1+x\right)^{n}=1+{\frac {n}{1!}}x+{\frac {n\left(n-1\right)}{2!}}x^{2}+\dots +{\frac {n\left(n-1\right)\dots \left(n-r+1\right)}{r!}}x^{r}+\dots } [ r ( cos θ + i sin θ ) ] n = r n ( cos n θ + i sin n θ ) {\displaystyle \left[r\left(\cos {\theta }+i\sin {\theta }\right)\right]^{n}=r^{n}\left(\cos {n\theta }+i\sin {n\theta }\right)} A − 1 = 1 det ( A ) adj ( A ) {\displaystyle A^{-1}={\frac {1}{\det \left(A\right)}}{\text{adj}}\left(A\right)} 等差數列
等比數列
∑ k = 1 n k = n ( n + 1 ) 2 {\displaystyle \sum _{k=1}^{n}k={\frac {n\left(n+1\right)}{2}}} ∑ k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 {\displaystyle \sum _{k=1}^{n}k^{2}={\frac {n\left(n+1\right)\left(2n+1\right)}{6}}} ∑ k = 1 n k 3 = [ n ( n + 1 ) 2 ] 2 {\displaystyle \sum _{k=1}^{n}k^{3}=\left[{\frac {n\left(n+1\right)}{2}}\right]^{2}}
弧長 = r θ {\displaystyle =r\theta } 扇形面積 = 1 2 r 2 θ {\displaystyle ={\frac {1}{2}}r^{2}\theta } tan θ = sin θ cos θ {\displaystyle \tan {\theta }={\frac {\sin {\theta }}{\cos {\theta }}}} csc θ = 1 sin θ {\displaystyle \csc {\theta }={\frac {1}{\sin {\theta }}}} sec θ = 1 cos θ {\displaystyle \sec {\theta }={\frac {1}{\cos {\theta }}}} cot θ = 1 tan θ {\displaystyle \cot {\theta }={\frac {1}{\tan {\theta }}}} a sin A = b sin B = c sin C = 2 R {\displaystyle {\frac {a}{\sin {A}}}={\frac {b}{\sin {B}}}={\frac {c}{\sin {C}}}=2R} a 2 = b 2 + c 2 − 2 b c cos A {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos {A}} Δ = 1 2 a b sin C {\displaystyle \Delta ={\frac {1}{2}}ab\sin {C}} Δ = s ( s − a ) ( s − b ) ( s − c ) , s = a + b + c 2 {\displaystyle \Delta ={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}},s={\frac {a+b+c}{2}}} 內切圓半徑 r = Δ s {\displaystyle r={\frac {\Delta }{s}}} sin 2 θ + cos 2 θ = 1 {\displaystyle \sin ^{2}{\theta }+\cos ^{2}{\theta }=1} sec 2 θ = 1 + tan 2 θ {\displaystyle \sec ^{2}{\theta }=1+\tan ^{2}{\theta }} csc 2 θ = 1 + cot 2 θ {\displaystyle \csc ^{2}{\theta }=1+\cot ^{2}{\theta }} sin ( A ± B ) = sin A cos B ± cos A sin B {\displaystyle \sin {\left(A\pm B\right)}=\sin {A}\cos {B}\pm \cos {A}\sin {B}} cos ( A ± B ) = cos A cos B ∓ sin A sin B {\displaystyle \cos {\left(A\pm B\right)}=\cos {A}\cos {B}\mp \sin {A}\sin {B}} tan ( A ± B ) = tan A ± tan B 1 ∓ tan A tan B {\displaystyle \tan {\left(A\pm B\right)}={\frac {\tan {A}\pm \tan {B}}{1\mp \tan {A}\tan {B}}}} sin 2 A = 2 sin A cos A {\displaystyle \sin {2A}=2\sin {A}\cos {A}} cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2 sin 2 A {\displaystyle {\begin{aligned}\cos {2A}&=\cos ^{2}{A}-\sin ^{2}{A}\\&=2\cos ^{2}{A}-1\\&=1-2\sin ^{2}{A}\\\end{aligned}}} tan 2 A = 2 tan A 1 − tan 2 A {\displaystyle \tan {2A}={\frac {2\tan {A}}{1-\tan ^{2}{A}}}} sin A cos B = sin ( A + B ) + sin ( A − B ) 2 {\displaystyle \sin {A}\cos {B}={\frac {\sin {\left(A+B\right)}+\sin {\left(A-B\right)}}{2}}} cos A cos B = cos ( A + B ) + cos ( A − B ) 2 {\displaystyle \cos {A}\cos {B}={\frac {\cos {\left(A+B\right)}+\cos {\left(A-B\right)}}{2}}} sin A sin B = cos ( A − B ) − cos ( A + B ) 2 {\displaystyle \sin {A}\sin {B}={\frac {\cos {\left(A-B\right)}-\cos {\left(A+B\right)}}{2}}} sin A + sin B = 2 sin A + B 2 cos A − B 2 {\displaystyle \sin {A}+\sin {B}=2\sin {\frac {A+B}{2}}\cos {\frac {A-B}{2}}} sin A − sin B = 2 cos A + B 2 sin A − B 2 {\displaystyle \sin {A}-\sin {B}=2\cos {\frac {A+B}{2}}\sin {\frac {A-B}{2}}} cos A + cos B = 2 cos A + B 2 cos A − B 2 {\displaystyle \cos {A}+\cos {B}=2\cos {\frac {A+B}{2}}\cos {\frac {A-B}{2}}} cos A − cos B = − 2 sin A + B 2 sin A − B 2 {\displaystyle \cos {A}-\cos {B}=-2\sin {\frac {A+B}{2}}\sin {\frac {A-B}{2}}}
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 {\displaystyle d={\sqrt {\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}} 分比公式 ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) {\displaystyle \left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}}\right)} 兩直線的夾角 θ {\displaystyle \theta } ， tan θ = | m 2 − m 1 1 + m 2 m 1 | {\displaystyle \tan {\theta }=\left|{\frac {m_{2}-m_{1}}{1+m_{2}m_{1}}}\right|} 直線方程是 y − y 1 = m ( x − x 1 ) {\displaystyle y-y_{1}=m\left(x-x_{1}\right)} 點到直線的距離 = | A x 0 + B y 0 + C A 2 + B 2 | {\displaystyle =\left|{\frac {Ax_{0}+By_{0}+C}{\sqrt {A^{2}+B^{2}}}}\right|} 三角形的面積 = 1 2 | ( x 1 y 2 + x 2 y 3 + x 3 y 1 ) − ( x 2 y 1 + x 3 y 2 + x 1 y 3 ) | {\displaystyle ={\frac {1}{2}}\left|\left(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}\right)-\left(x_{2}y_{1}+x_{3}y_{2}+x_{1}y_{3}\right)\right|} 圓的標準式 ( x − h ) 2 + ( y − k ) 2 = r 2 {\displaystyle \left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}} 平移 { x = x ′ + h y = y ′ + k {\displaystyle {\begin{cases}x=x^{\prime }+h\\y=y^{\prime }+k\\\end{cases}}} 轉軸 { x = x ′ cos θ − y ′ sin θ y = y ′ sin θ + y ′ cos θ {\displaystyle {\begin{cases}x=x^{\prime }\cos {\theta }-y^{\prime }\sin {\theta }\\y=y^{\prime }\sin {\theta }+y^{\prime }\cos {\theta }\\\end{cases}}} 拋物線
橢圓
雙曲線
平均數 x ¯ = ∑ f i x i ∑ f i {\displaystyle {\bar {x}}={\frac {\sum {f_{i}x_{i}}}{\sum {f_{i}}}}} 平均差 = ∑ | x i − x ¯ | f i ∑ f i {\displaystyle ={\frac {\sum {\left|x_{i}-{\bar {x}}\right|f_{i}}}{\sum {f_{i}}}}} 中位數 M = L + ( n 2 − F m f m ) C m {\displaystyle M=L+\left({\frac {{\frac {n}{2}}-F_{m}}{f_{m}}}\right)C_{m}} 眾數 = L + ( d 1 d 1 + d 2 ) C {\displaystyle =L+\left({\frac {d_{1}}{d_{1}+d{2}}}\right)C} 上四分位數 Q 3 = L 3 + ( 3 n 4 − F 3 f 3 ) C 3 {\displaystyle Q_{3}=L{3}+\left({\frac {{\frac {3n}{4}}-F_{3}}{f_{3}}}\right)C_{3}} 下四分位數 Q 1 = L 1 + ( n 4 − F 1 f 1 ) C 1 {\displaystyle Q_{1}=L{1}+\left({\frac {{\frac {n}{4}}-F_{1}}{f_{1}}}\right)C_{1}} 四分位距 = Q 3 − Q 1 {\displaystyle =Q_{3}-Q_{1}} 四分位差 Q . D . = Q 3 − Q 1 2 {\displaystyle Q.D.={\frac {Q_{3}-Q_{1}}{2}}} 方差 σ 2 = ∑ ( x i − x ¯ ) 2 f i ∑ f i = ∑ x i 2 f i ∑ f i − x ¯ 2 {\displaystyle \sigma ^{2}={\frac {\sum {\left(x_{i}-{\bar {x}}\right)^{2}f_{i}}}{\sum {f_{i}}}}={\frac {\sum {x_{i}^{2}f_{i}}}{\sum {f_{i}}}}-{\bar {x}}^{2}} 標準差 σ = ∑ ( x i − x ¯ ) 2 f i ∑ f i = ∑ x i 2 f i ∑ f i − x ¯ 2 {\displaystyle \sigma ={\sqrt {\frac {\sum {\left(x_{i}-{\bar {x}}\right)^{2}f_{i}}}{\sum {f_{i}}}}}={\sqrt {{\frac {\sum {x_{i}^{2}f_{i}}}{\sum {f_{i}}}}-{\bar {x}}^{2}}}} 統計指數 I = Q 1 Q 0 × 100 {\displaystyle I={\frac {Q_{1}}{Q_{0}}}\times 100} 綜合指數 = ∑ w i x i ∑ w i {\displaystyle ={\frac {\sum {w_{i}x_{i}}}{\sum {w_{i}}}}} P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) {\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)} P ( A ) = 1 − P ( A ′ ) {\displaystyle P\left(A\right)=1-P\left(A^{\prime }\right)} n P r = n ! ( n − r ) ! {\displaystyle {_{n}}P_{r}={\frac {n!}{\left(n-r\right)!}}} n C r = n ! ( n − r ) ! r ! {\displaystyle {_{n}}C_{r}={\frac {n!}{\left(n-r\right)!r!}}} 期望值 E = x 1 p 1 + x 2 p 2 + ⋯ + x k p k {\displaystyle E=x_{1}p_{1}+x_{2}p_{2}+\dots +x_{k}p_{k}} 二項分配 P ( X = r ) = n C r p r q n − r {\displaystyle P\left(X=r\right)={_{n}}C_{r}p^{r}q^{n-r}}
lim x → 0 sin x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin {x}}{x}}=1} lim x → ∞ ( 1 + 1 x ) x = e {\displaystyle \lim _{x\to \infty }{\left(1+{\frac {1}{x}}\right)^{x}}=e} d d x ( u v ) = u d v d x + v d u d x {\displaystyle {\frac {d}{dx}}\left(uv\right)=u{\frac {dv}{dx}}+v{\frac {du}{dx}}} d d x ( u v ) = v d u d x − u d v d x v 2 {\displaystyle {\frac {d}{dx}}\left({\frac {u}{v}}\right)={\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}} d y d x = d y d u d u d x {\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}{\frac {du}{dx}}} d d x f ( g ( x ) ) = f ′ ( g ( x ) ) g ′ ( x ) {\displaystyle {\frac {d}{dx}}f\left(g\left(x\right)\right)=f^{\prime }\left(g\left(x\right)\right)g^{\prime }\left(x\right)} d d x x n = n x x − 1 {\displaystyle {\frac {d}{dx}}x^{n}=nx^{x-1}} d d x sin x = cos x {\displaystyle {\frac {d}{dx}}\sin {x}=\cos {x}} d d x cos x = − sin x {\displaystyle {\frac {d}{dx}}\cos {x}=-\sin {x}} d d x tan x = sec 2 x {\displaystyle {\frac {d}{dx}}\tan {x}=\sec ^{2}{x}} d d x cot x = − csc 2 x {\displaystyle {\frac {d}{dx}}\cot {x}=-\csc ^{2}{x}} d d x sec x = sec x tan x {\displaystyle {\frac {d}{dx}}\sec {x}=\sec {x}\tan {x}} d d x csc x = − csc x cot x {\displaystyle {\frac {d}{dx}}\csc {x}=-\csc {x}\cot {x}} d d x ln x = 1 x {\displaystyle {\frac {d}{dx}}\ln {x}={\frac {1}{x}}} d d x log a x = 1 x ln a {\displaystyle {\frac {d}{dx}}\log _{a}x={\frac {1}{x\ln {a}}}} d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}} d d x a x = a x ln a {\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln {a}} d d x sin − 1 x = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\sin ^{-1}{x}={\frac {1}{\sqrt {1-x^{2}}}}} d d x tan − 1 x = 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\tan ^{-1}{x}={\frac {1}{1+x^{2}}}} ∫ x n d x = x n + 1 n + 1 + C , n ≠ − 1 {\displaystyle \int x^{n}dx={\frac {x^{n+1}}{n+1}}+C,n\neq -1} ∫ cos x d x = s i n x + C {\displaystyle \int \cos {x}dx=sin{x}+C} ∫ sin x d x = − cos x + C {\displaystyle \int \sin {x}dx=-\cos {x}+C} ∫ sec 2 x d x = tan x + C {\displaystyle \int \sec ^{2}{x}dx=\tan {x}+C} ∫ csc 2 x d x = cot x + C {\displaystyle \int \csc ^{2}{x}dx=\cot {x}+C} ∫ sec x tan x d x = sec x + C {\displaystyle \int \sec {x}\tan {x}dx=\sec {x}+C} ∫ csc x cot x d x = − csc x + C {\displaystyle \int \csc {x}\cot {x}dx=-\csc {x}+C} ∫ 1 x d x = ln x + C {\displaystyle \int {\frac {1}{x}}dx=\ln {x}+C} ∫ e x d x = e x + C {\displaystyle \int e^{x}dx=e^{x}+C} ∫ a x d x = a x ln a + C {\displaystyle \int a^{x}dx={\frac {a^{x}}{\ln {a}}}+C} ∫ d x 1 − x 2 = sin − 1 x + C {\displaystyle \int {\frac {dx}{\sqrt {1-x^{2}}}}=\sin ^{-1}{x}+C} ∫ d x 1 + x 2 = tan − 1 x + C {\displaystyle \int {\frac {dx}{1+x^{2}}}=\tan ^{-1}{x}+C} 面積 ∫ a b y d x {\displaystyle \int _{a}^{b}ydx} 或 ∫ a b x d y {\displaystyle \int _{a}^{b}xdy} 或 ∫ α β 1 2 [ r ( θ ) ] 2 d θ {\displaystyle \int _{\alpha }^{\beta }{\frac {1}{2}}\left[r\left(\theta \right)\right]^{2}d\theta } 體積 π ∫ a b y 2 d x {\displaystyle \pi \int _{a}^{b}y^{2}dx} 或 π ∫ a b x 2 d y {\displaystyle \pi \int _{a}^{b}x^{2}dy} 牛頓法 x n = x n − 1 = f ( x n − 1 ) f ′ ( x n − 1 ) {\displaystyle x_{n}=x_{n-1}={\frac {f\left(x_{n-1}\right)}{f^{\prime }\left(x_{n-1}\right)}}} 梯形法 ∫ a b f ( x ) d x ≈ b − a n ( y 0 + y n 2 + y 1 + y 2 + ⋯ + y n − 1 ) {\displaystyle \int _{a}^{b}f\left(x\right)dx\approx {\frac {b-a}{n}}\left({\frac {y_{0}+y_{n}}{2}}+y_{1}+y_{2}+\dots +y_{n-1}\right)} 辛普遜法 ∫ a b f ( x ) d x ≈ b − a 6 n [ ( y 0 + y 2 n ) + 4 ( y 1 + y 3 + ⋯ + y 2 n − 1 ) + 2 ( y 2 + y 4 + ⋯ + y 2 n − 2 ) ] {\displaystyle \int _{a}^{b}f\left(x\right)dx\approx {\frac {b-a}{6n}}\left[\left(y_{0}+y_{2n}\right)+4\left(y_{1}+y_{3}+\dots +y_{2n-1}\right)+2\left(y_{2}+y_{4}+\dots +y_{2n-2}\right)\right]}
![{\displaystyle \left[r\left(\cos {\theta }+i\sin {\theta }\right)\right]^{n}=r^{n}\left(\cos {n\theta }+i\sin {n\theta }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10066ef7e351e077c55e08f0af2b659095bc7c3b)
![{\displaystyle S_{n}={\frac {n}{2}}\left[2a+\left(n-1\right)d\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a41bc9111209f6863b31d836c3692ed1f031a0bf)
![{\displaystyle \sum _{k=1}^{n}k^{3}=\left[{\frac {n\left(n+1\right)}{2}}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8029449361675d9166b08c08a17e5fadfb6443)
，
或
或![{\displaystyle \int _{\alpha }^{\beta }{\frac {1}{2}}\left[r\left(\theta \right)\right]^{2}d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/26d6b25564ce82f20f4c078759a57ed6140b39f6)
或
![{\displaystyle \int _{a}^{b}f\left(x\right)dx\approx {\frac {b-a}{6n}}\left[\left(y_{0}+y_{2n}\right)+4\left(y_{1}+y_{3}+\dots +y_{2n-1}\right)+2\left(y_{2}+y_{4}+\dots +y_{2n-2}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3359bdc6031defd22661505c23ae1dbfd98d910e)
