Mathematics Formulas & Tables

$x^2$

1. $ax^2+bx+c=0(a\neq0),x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
2. $x^2+y^2=r^2(r>0)$
3. $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1(a\geq b,c=\sqrt{a^2-b^2})$
4. $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1(a\leq b,c=\sqrt{b^2-a^2})$
5. $y^2=2px(p>0)$
6. $(a\pm b)^2=a^2\pm2ab+b^2$
7. $(a+b)(a-b)=a^2-b^2$
8. $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$
9. $a^2+b^2\geq2ab$
10. $\dfrac{a+b}2\geq\sqrt{ab}(a\geq0,b\geq0)$

$x^3$

1. $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)=(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2]$
2. $(a\pm b)^3=a^3\pm3a^2b+3ab^2\pm b^3$
3. $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$

$x^n$

1. $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+...+a^{n-k}b^k+...+b^{n-1})$
2. $a^{2n+1}+b^{2n+1}=(a+b)[a^{2n}-a^{2n-1}b+...+(-1)^ka^{n-k}b^k+...+b^{2n}]$
3. $(a+b)^n=C_n^0a^n+C_n^1a^{n-1}b+...+C_n^ka^{n-k}b^k+...+C_n^nb^n$
4. $(a-b)^n=C_n^0a^n-C_n^1a^{n-1}b+...+(-1)^kC_n^ka^{n-k}b^k+...+(-1)^nC_n^nb^n$

Inequalities

1. Mean value inequality: If $a_1,a_2,...,a_n\geq0$, then $\dfrac n{\frac1{a_1}+\frac1{a_2}+...+\frac1{a_n}}\leq\sqrt[n]{a_1a_2...a_n}\leq\dfrac{a_1+a_2+...+a_n}n\leq\sqrt{\dfrac{a_1^2+a_2^2+...+a_n^2}n}$, the equal sign holds iff $a_1=a_2=...=a_n$
2. Power mean inequality: If $a_1,a_2,...,a_n\geq0,\alpha>\beta>0$, then $\left(\dfrac{a_1^\alpha+a_2^\alpha+...+a_n^\alpha}n\right)^{\frac1\alpha}\geq\left(\dfrac{a_1^\beta+a_2^\beta+...+a_n^\beta}n\right)^{\frac1\beta}$, the equal sign holds iff $a_1=a_2=...=a_n$
3. Weighted mean value inequality: If $a_1,a_2,...,a_n,\alpha_i>0,\sum_{i=1}^n\alpha_i=1$, then $a_1^{\alpha_1}a_2^{\alpha_2}...a_n^{\alpha_n}\leq a_1\alpha_1+a_2\alpha_2+...+a_n\alpha_n$
4. Cauchy's inequality: $(a_1b_1+a_2b_2+...+a_nb_n)^2\leq(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)$, the equal sign holds iff $b_1=b_2=...=b_n=0$ or $\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=...=\dfrac{a_n}{b_n}$
5. Sorting inequality: If $a_1\leq a_2\leq...\leq a_n,b_1\leq b_2\leq...\leq b_n$, then $a_1b_n+a_2b_{n-1}+...+a_nb_1\leq a_1b_{t_1}+a_2b_{t_2}+...+a_nb_{t_n}\leq a_1b_1+a_2b_2+...+a_nb_n(\{t_1,t_2,...,t_n\}=\{1,2,...,n\})$
6. Chebyshev's inequality: If $a_1\leq a_2\leq...\leq a_n,b_1\leq b_2\leq...\leq b_n$, then $n\sum_{i=1}^n a_kb_{n-k+1}\leq\sum_{i=1}^n a_k\sum_{i=1}^n b_k\leq n\sum_{i=1}^n a_kb_k$
7. Jensen's inequality: For a concave function $f(x),x\in(a,b),x_1,x_2,...,x_n\in(a,b)$, then $f(\dfrac{\sum_{i=1}^n a_ix_i}{\sum_{i=1}^n a_i}) \geq\dfrac{\sum_{i=1}^n a_if(x_i)}{\sum_{i=1}^n a_i}$
8. Schur's inequality: If $x,y,z\geq0$, then $x^r(x-y)(x-z)+y^r(y-z)(y-x)+z^r(z-x)(z-y)\geq0$
9. Hölder's inequality: If $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p,q>0,\frac1p+\frac1q=1$, then $\sum_{i=1}^n a_ib_i\leq(\sum_{i=1}^n a_i^p)^{\frac1p}(\sum_{i=1}^n b_i^q)^{\frac1q}$, the equal sign holds iff $b_1=b_2=...=b_n=0$ or $\dfrac{a_1^p}{b_1^q}=\dfrac{a_2^p}{b_2^q}=...=\dfrac{a_n^p}{b_n^q}$
10. If $a_1,a_2,...,a_n,b_1,b_2,...,b_n,m>0$, then $\sum_{i=1}^n\dfrac{a_i^{m+1}}{b_i^m}\geq\dfrac{(\sum_{i=1}^n a_i)^{m+1}}{(\sum_{i=1}^n b_i)^m}$, the equal sign holds iff $\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=...=\dfrac{a_n}{b_n}$
11. Carlson inequality: If $a_{ij},\alpha_i\geq0,\sum_{i=1}^n\alpha_i=1$, then $\prod_{j=1}^m(\sum_{i=1}^n a_{ij})^{\alpha_j}\geq\sum_{i=1}^n(\prod_{j=1}^m a_{ij}^{\alpha_j})$
12. Abel sum: Let $S_k=\sum_{i=1}^k a_i$, then $\sum_{i=1}^n a_kb_k=\sum_{i=1}^{n-1}S_i(b_i-b_{i+1})+S_nb_n$

Trigonometric functions

1. $\sin a\csc a=1$
2. $\cos a\sec a=1$
3. $\tan a\cot a=1$
4. $\dfrac{\sin a}{\cos a}=\tan a$
5. $\sin^2a+\cos^2a=1$
6. $\tan^2a+1=\sec^2a$
7. $\cot^2a+1=\csc^2a$
8. $\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b$
9. $\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b$
10. $\tan(a\pm b)=\dfrac{\tan a\pm\tan b}{1\mp\tan a\tan b}$
11. $\sin2a=2\sin a\cos a$
12. $\cos2a=\cos^2a-\sin^2a=1-2\sin^2a=2\cos^2a-1$
13. $\tan2a=\dfrac{2\tan a}{1-\tan^2a}$
14. $\sin\dfrac a2=\pm\sqrt{\dfrac{1-\cos a}2}$
15. $\cos\dfrac a2=\pm\sqrt{\dfrac{1+\cos a}2}$
16. $\tan\dfrac a2=\pm\sqrt{\dfrac{1-\cos a}{1+\cos a}}=\dfrac{\sin a}{1+\cos a}=\dfrac{1-\cos a}{\sin a}$
17. $\sin a+\sin b=2\sin\dfrac{a+b}2\cos\dfrac{a-b}2$
18. $\sin a-\sin b=2\cos\dfrac{a+b}2\sin\dfrac{a-b}2$
19. $\cos a+\cos b=2\cos\dfrac{a+b}2\cos\dfrac{a-b}2$
20. $\cos a-\cos b=-2\sin\dfrac{a+b}2\sin\dfrac{a-b}2$
21. $\tan a\pm\tan b=\dfrac{\sin(a\pm b)}{\cos a\cos b}=\tan(a\pm b)\mp\tan a\tan b\tan(a\pm b)$
22. $\sin a\cos b=\dfrac12[\sin(a+b)+\sin(a-b)]$
23. $\cos a\cos b=\dfrac12[\cos(a+b)+\cos(a-b)]$
24. $\sin a\sin b=-\dfrac12[\cos(a+b)-\cos(a-b)]$
25. $\sin3a=3\sin a-4\sin^3a$
26. $\cos3a=4\cos^3a-3\cos a$
27. $\tan3a=\tan a\tan(60-a)\tan(60+a)$
28. $\sin a=\dfrac{2\tan\dfrac a2}{1+\tan^2\dfrac a2}$
29. $\cos a=\dfrac{1-\tan^2\dfrac a2}{1+\tan^2\dfrac a2}$
30. $\tan a=\dfrac{2\tan\dfrac a2}{1-\tan^2\dfrac a2}$
31. $\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}=2R$
32. $c^2=a^2+b^2-2ab\cos C$
33. $\sin A+\sin B+\sin C=4\cos{\dfrac A2}\cos{\dfrac B2}\cos{\dfrac C2}$
34. $\cos A+\cos B+\cos C=1+4\sin{\dfrac A2}\sin{\dfrac B2}\sin{\dfrac C2}$
35. $\sin^2A+\sin^2B+\sin^2C=2+2\cos A\cos B\cos C$
36. $\cos^2A+\cos^2B+\cos^2C=1-2\cos A\cos B\cos C$
37. $\sin^2{\dfrac A2}+\sin^2{\dfrac B2}+\sin^2{\dfrac C2}=1-2\sin{\dfrac A2}\sin{\dfrac B2}\sin{\dfrac C2}$
38. $\cos^2{\dfrac A2}+\cos^2{\dfrac B2}+\cos^2{\dfrac C2}=2+2\sin{\dfrac A2}\sin{\dfrac B2}\sin{\dfrac C2}$
39. $\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C$
40. $\cos2A+\cos2B+\cos2C=-1-4\cos A\cos B\cos C$
41. $\tan A+\tan B+\tan C=\tan A\tan B\tan C$
42. $\cot A\cot B+\cot A\cot C+\cot B\cot C=1$
43. $\cot{\dfrac A2}+\cot{\dfrac B2}+\cot{\dfrac C2}=\cot{\dfrac A2}\cot{\dfrac B2}\cot{\dfrac C2}$
44. $\tan{\dfrac A2}\tan{\dfrac B2}+\tan{\dfrac A2}\tan{\dfrac C2}+\tan{\dfrac B2}\tan{\dfrac C2}=1$

Inverse trigonometric functions

1. $\arcsin a+\arccos a=\dfrac\pi2$
2. $\arctan a+\operatorname{arccot}a=\dfrac\pi2$
3. $\cos\arcsin a=\sin\arccos a$
4. $\arcsin\cos a=\arccos\sin a\;(2k\pi\leq a\leq 2k\pi+\dfrac\pi2)$
5. $\arctan a+\arctan b=\arctan\dfrac{a+b}{1-ab}+k\pi$

Limits and calculus

1. $\lim_{x\rightarrow0}\dfrac{\sin x}x=1$
2. $\lim_{x\rightarrow\infty}(1+\dfrac1x)^x=e$
3. $\lim_{x\rightarrow x_0}(f(x)\pm g(x))=\lim_{x\rightarrow x_0}f(x)\pm\lim_{x\rightarrow x_0}g(x)$
4. $\lim_{x\rightarrow x_0}(f(x)g(x))=\lim_{x\rightarrow x_0}f(x)\times\lim_{x\rightarrow x_0}g(x)$
5. $\lim_{x\rightarrow x_0}\dfrac{f(x)}{g(x)}=\dfrac{\lim_{x\rightarrow x_0}f(x)}{\lim_{x\rightarrow x_0}g(x)}(\lim_{x\rightarrow x_0}g(x)\neq0)$
6. $\lim_{x\rightarrow x_0}f(x)^n=[\lim_{x\rightarrow x_0}f(x)]^n$
7. $a'=0$
8. $(x^a)'=ax^{a-1}(a\neq0)$
9. $(a^x)'=a^x\ln a$
10. $(e^x)'=e^x$
11. $(\log_ax)'=\dfrac1x\log_ae$
12. $(\ln x)'=\dfrac1x$
13. $(\sin x)'=\cos x$
14. $(\cos x)'=-\sin x$
15. $(\tan x)'=\sec^2x$
16. $(\cot x)'=-\csc^2x$
17. $(\sec x)'=\sec x\tan x$
18. $(\csc x)'=-\csc x\cot x$
19. $(\arcsin x)'=\dfrac1{\sqrt{1-x^2}}$
20. $(\arccos x)'=-\dfrac1{\sqrt{1-x^2}}$
21. $(\arctan x)'=\dfrac1{1+x^2}$
22. $(\operatorname{arccot}x)'=-\dfrac1{1-x^2}$
23. $(f(x)\pm g(x))'=f'(x)\pm g'(x)$
24. $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$
25. $(\dfrac{f(x)}{g(x)})'=\dfrac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$
26. $\int0dx=C$
27. $\int adx=ax+C$
28. $\int x^adx=\dfrac1{a+1}x^{a+1}+C(a\neq-1)$
29. $\int a^xdx=a^x\log_ae+C$
30. $\int e^xdx=e^x+C$
31. $\int \dfrac1xdx=\ln|x|+C$
32. $\int\sin xdx=-\cos x+C$
33. $\int\cos xdx=\sin x+C$
34. $\int(f(x)\pm g(x))dx=\int f(x)dx\pm\int g(x)dx$

Summations of series

1. $\sum_{n=1}^xn=\dfrac12x(x+1)$
2. $\sum_{n=1}^xn^2=\dfrac16x(x+1)(2x+1)$
3. $\sum_{n=1}^xn^3=\dfrac14x^2(x+1)^2$
4. $\sum_{n=0}^xa^n=\dfrac{a^{x+1}-1}{a-1}$
5. $\sum_{n=0}^x2^n=2^{x+1}-1$
6. $\sum_{n=1}^\infty\dfrac1{a^n}=\dfrac1{a-1}(|a|>1)$
7. $\sum_{n=1}^\infty\dfrac1{2^n}=1$
8. $\sum_{n=1}^\infty\dfrac1{n^2}=\dfrac{\pi^2}6$
9. $\sum_{n=1}^\infty\dfrac1{n^4}=\dfrac{\pi^4}{90}$
10. $\sum_{n=0}^\infty\dfrac{a^n}{n!}=e^a$
11. $\sum_{n=0}^\infty\dfrac{(-1)^na^{2n+1}}{(2n+1)!}=\sin a$
12. $\sum_{n=0}^\infty\dfrac{(-1)^na^{2n}}{(2n)!}=\cos a$

Perimeters and areas of plane figures

Surface areas and volumes of solid figures

Power of $2$

Power of $3$

Power of $5$

Frequently used pythagorean numbers (simplest)

1. 3, 4, 5
2. 5, 12, 13
3. 7, 24, 25
4. 8, 15, 17
5. 9, 40, 41
6. 12, 35, 37
7. 20, 21, 29

Frequently used triangle with 120 degrees (simplest)

1. 3, 5, 7
2. 7, 8, 13
3. 5, 16, 19
4. 8, 18, 23

Frequently used triangle with 60 degrees (simplest)

1. 1, 1, 1
2. 3, 7, 8
3. 5, 7, 8
4. 5, 19, 21
5. 7, 13, 15
6. 8, 13, 15
7. 8, 23, 26
8. 16, 19, 21
9. 18, 23, 26

$\pi$

$\pi\approx3.$

$5820974944\,5923078164\,0628620899\,8628034825\,3421170679$

$8214808651\,3282306647\,0938446095\,5058223172\,5359408128$

$4811174502\,8410270193\,8521105559\,6446229489\,5493038196$

$4428810975\,6659334461\,2847564823\,3786783165\,2712019091$

$4564856692\,3460348610\,4543266482\,1339360726\,0249141273$

$7245870066\,0631558817\,4881520920\,9628292540\,9171536436$

$7892590360\,0113305305\,4882046652\,1384146951\,9415116094$

$3305727036\,5759591953\,0921861173\,8193261179\,3105118548$

$0744623799\,6274956735\,1885752724\,8912279381\,8301194912$

$9833673362\,4406566430\,8602139494\,6395224737\,1907021798$

$6094370277\,0539217176\,2931767523\,8467481846\,7669405132$

$0005681271\,4526356082\,7785771342\,7577896091\,7363717872$

$1468440901\,2249534301\,4654958537\,1050792279\,6892589235$

$4201995611\,2129021960\,8640344181\,5981362977\,4771309960$

$5187072113\,4999999837\,2978049951\,0597317328\,1609631859$

$5024459455\,3469083026\,4252230825\,3344685035\,2619311881$

$7101000313\,7838752886\,5875332083\,8142061717\,7669147303$

$5982534904\,2875546873\,1159562863\,8823537875\,9375195778$

$1857780532\,1712268066\,1300192787\,6611195909\,2164201989$

$\tau$

$\tau\approx6.$

$1641949889\,1846156328\,1257241799\,7256069650\,6842341359$

$6429617302\,6564613294\,1876892191\,0116446345\,0718816256$

$9622349005\,6820540387\,7042211119\,2892458979\,0986076392$

$8857621951\,3318668922\,5695129646\,7573566330\,5424038182$

$9129713384\,6920697220\,9086532964\,2678721452\,0498282547$

$4491740132\,1263117634\,9763041841\,9256585081\,8343072873$

$5785180720\,0226610610\,9764093304\,2768293903\,8830232188$

$6611454073\,1519183906\,1843722347\,6386522358\,6210237096$

$1489247599\,2549913470\,3771505449\,7824558763\,6602389825$

$9667346724\,8813132861\,7204278989\,2790449474\,3814043597$

$2188740554\,1078434352\,5863535047\,6934963693\,5338810264$

$0011362542\,9052712165\,5571542685\,5155792183\,4727435744$

$2936881802\,4499068602\,9309917074\,2101584559\,3785178470$

$8403991222\,4258043921\,7280688363\,1962725954\,9542619921$

$0374144226\,9999999674\,5956099902\,1194634656\,3219263719$

$0048918910\,6938166052\,8504461650\,6689370070\,5238623763$

$4202000627\,5677505773\,1750664167\,6284123435\,5338294607$

$1965069808\,5751093746\,2319125727\,7647075751\,8750391556$

$3715561064\,3424536132\,2600385575\,3222391818\,4328403978$

$e$

$e\approx2.$

$9574966967\,6277240766\,3035354759\,4571382178\,5251664274$

$2746639193\,2003059921\,8174135966\,2904357290\,0334295260$

$5956307381\,3232862794\,3490763233\,8298807531\,9525101901$

$1573834187\,9307021540\,8914993488\,4167509244\,7614606680$

$8226480016\,8477411853\,7423454424\,3710753907\,7744992069$

$5517027618\,3860626133\,1384583000\,7520449338\,2656029760$

$6737113200\,7093287091\,2744374704\,7230696977\,2093101416$

$9283681902\,5515108657\,4637721112\,5238978442\,5056953696$

$7707854499\,6996794686\,4454905987\,9316368892\,3009879312$

$7736178215\,4249992295\,7635148220\,8269895193\,6680331825$

$2886939849\,6465105820\,9392398294\,8879332036\,2509443117$

$3012381970\,6841614039\,7019837679\,3206832823\,7646480429$

$5311802328\,7825098194\,5581530175\,6717361332\,0698112509$

$9618188159\,3041690351\,5988885193\,4580727386\,6738589422$

$8792284998\,9208680582\,5749279610\,4841984443\,6346324496$

$8487560233\,6248270419\,7862320900\,2160990235\,3043699418$

$4914631409\,3431738143\,6405462531\,5209618369\,0888707016$

$7683964243\,7814059271\,4563549061\,3031072085\,1038375051$

$0115747704\,1718986106\,8739696552\,1267154688\,9570350354$

$Ð$

$Ð\approx0.$

$9281784197\,9107908364\,5819149637\,2089369026\,1711409559$

$8621679226\,7152527586\,6935894686\,2618252825\,4561281924$

$9647991571\,1991841866\,9863599724\,5221416574\,4205050207$

$7784686934\,5390733422\,5050978540\,5274496866\,4095158612$

$7583776330\,9458526519\,9627816965\,9131567494\,8719351465$

$2455794583\,7982911446\,4$

Table of trigonometric functions

Download link of the next table (exsct.xlsx).