386 Formule de matematică disponibile

$\sin^2 x + \cos^2 x = 1$

$\sin(\frac{\pi}{2} - x) = \cos x$

$\sin 2x = 2 \sin x \cos x$

$\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$

$\tg 2x = \frac{2 \tg x}{1 - \tg^2 x}$

$\sin^2 x = \frac{1 - \cos 2x}{2}$

$\cos^2 x = \frac{1 + \cos 2x}{2}$

$\sin 3x = 4\sin^3 x - 3\sin x$

$\cos 3x = 4\cos^3 x - 3\cos x$

$\sin(a + b) = \sin a \cos b + \cos a \sin b$

$\cos(a + b) = \cos a \cos b - \sin a \sin b$

$\sin(a - b) = \sin a \cos b - \cos a \sin b$

$\cos(a - b) = \cos a \cos b + \sin a \sin b$

$\tg(a \pm b) = \frac{\tg a \pm \tg b}{1 \mp \tg a \cdot \tg b}$

$\tg \frac{a}{2} = \frac{1 - \cos a}{\sin a}$

$\left|\cos \frac{a}{2}\right| = \sqrt{\frac{1 + \cos a}{2}}$

$\left|\sin \frac{a}{2}\right| = \sqrt{\frac{1 - \cos a}{2}}$

$\sin a \cdot \cos b = \frac{1}{2}[\sin(a+b) + \sin(a-b)]$

$\cos a \cdot \cos b = \frac{1}{2}[\cos(a+b) + \cos(a-b)]$

$\sin a \cdot \sin b = -\frac{1}{2}[\cos(a+b) - \cos(a-b)]$

$\cos a + \cos b = 2\cos \frac{a+b}{2} \cos \frac{a-b}{2}$

$\cos a - \cos b = -2\sin \frac{a+b}{2} \sin \frac{a-b}{2}$

$\sin a + \sin b = 2\sin \frac{a+b}{2} \cos \frac{a-b}{2}$

$\sin a - \sin b = 2\sin \frac{a-b}{2} \cos \frac{a+b}{2}$

$\tg a + \tg b = \frac{\sin(a+b)}{\cos a \cos b}$

$\tg a - \tg b = \frac{\sin(a-b)}{\cos a \cos b}$

$\sin a = \frac{2\tg \frac{a}{2}}{1 + \tg^2 \frac{a}{2}}$

$\cos a = \frac{1 - \tg^2 \frac{a}{2}}{1 + \tg^2 \frac{a}{2}}$

$\tg a = \frac{2\tg \frac{a}{2}}{1 - \tg^2 \frac{a}{2}}$

$(a \pm b)^2 = a^2 \pm 2ab + b^2$

$(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3$

$a^2 - b^2 = (a - b)(a + b)$

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

$a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \ldots + b^{n-1})$

$a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + \ldots + b^{n-1})$

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$

$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)$

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

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

$\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2$

$\lbrack x \rbrack$

$\{ x \} = x - \lbrack x \rbrack$

$\frac{a}{b} + \frac{b}{a} \geq 2$

$x \cdot y \leq \left( \frac{x + y}{2} \right)^2$

$|x|$

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

$\int x^a dx = \frac{x^{a+1}}{a+1} + C$

$\int a^x dx = \frac{a^x}{\ln a} + C$

$\int \frac{1}{x} dx = \ln |x| + C$

$\int \frac{1}{x^2-a^2} dx = \frac{1}{2a} \ln |\frac{x-a}{x+a}| + C$

$\int \frac{1}{x^2+a^2} dx = \frac{1}{a} \arctg \frac{x}{a} + C$

$\int \sin x dx = -\cos x + C$

$\int \cos x dx = \sin x + C$

$\int \frac{1}{\cos^2 x} dx = \tg x + C$

$\int \frac{1}{\sin^2 x} dx = -\ctg x + C$

$\int \tg x dx = -\ln|\cos x| + C$

$\int \ctg x dx = \ln|\sin x| + C$

$\int \frac{1}{\sqrt{x^2+a^2}} dx = \ln|x + \sqrt{x^2+a^2}| + C$

$\int \frac{1}{\sqrt{x^2-a^2}} dx = \ln|x + \sqrt{x^2-a^2}| + C$

$\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin \frac{x}{a} + C$

$\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2} \arcsin \frac{x}{a} + C$

$\int \sqrt{x^2+a^2} dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2} \ln|x + \sqrt{x^2+a^2}| + C$

$\int \sqrt{x^2-a^2} dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2} \ln|x + \sqrt{x^2-a^2}| + C$

$\frac{BA'}{A'C} \cdot \frac{CB'}{B'A} \cdot \frac{AC'}{C'B} = 1$

$\overrightarrow{AD}(b+c) = b\overrightarrow{AB} + c\overrightarrow{AC}$

$\overrightarrow{HA} + \overrightarrow{HB} + \overrightarrow{HC} = 2\overrightarrow{HO}$

$\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{OH}$

$\|\vec{u}\| = \sqrt{a^2 + b^2}$

$\|\vec{v}\| = \sqrt{x^2 + y^2 + z^2}$

$\vec{v_1} \cdot \vec{v_2} = x_1x_2 + y_1y_2 + z_1z_2$

$f'(c) = 0$

$f'(c) = 0, c \in (a, b)$

$\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}$

$f(b) - f(a) = (b - a) \cdot f'(c)$

$\lim_{x \to b} \frac{f(x)}{g(x)} = \lim_{x \to b} \frac{f'(x)}{g'(x)} = l$

$f'(x) > 0$

$f'(x) < 0$

$l'_s \neq l'_d, \text{cel puțin una finită}$

$f'_s(x_0) = \pm\infty, f'_d(x_0) = \mp\infty$

$\lim_{x \to \pm\infty} f(x) = y_0$

$\lim_{x \to x_0} f(x) = \pm\infty$

$\lim_{x \to \pm\infty} [f(x) - (mx + n)] = 0$

$(c)' = 0$

$(x)' = 1$

$(x^n)' = n \cdot x^{n-1}$

$(x^r)' = r \cdot x^{r-1}$

$(\sqrt{x})' = \frac{1}{2\sqrt{x}}$

$(\ln x)' = \frac{1}{x}$

$(e^x)' = e^x$

$(a^x)' = a^x \cdot \ln a$

$(\sin x)' = \cos x$

$(\cos x)' = -\sin x$

$(\tg x)' = \frac{1}{\cos^2 x}$

$(\ctg x)' = -\frac{1}{\sin^2 x}$

$(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}$

$(\arccos x)' = -\frac{1}{\sqrt{1-x^2}}$

$(\arctg x)' = \frac{1}{1+x^2}$

$(\arcctg x)' = -\frac{1}{1+x^2}$

$(\log_a x)' = \frac{1}{x \ln a}$

$(u^v)' = u^v \cdot (v' \cdot \ln u + v \cdot \frac{u'}{u})$

$\log_a x = y \iff a^y = x$

$a^{\log_a x} = x$

$\log_a x^k = k \cdot \log_a x$

$\log_a (xy) = \log_a x + \log_a y$

$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

$a_{n+1} = a_n + r$

$a_n = a_1 + (n-1)r$

$S_n = \frac{(a_1 + a_n) \cdot n}{2}$

$b_n = b_1 \cdot q^{n-1}$

$S_n = \frac{b_1(q^n - 1)}{q - 1}, q \neq 1$

$f = a_nX^n + a_{n-1}X^{n-1} + ... + a_1X + a_0$

$grad(f + g) = grad\ f + grad\ g$

$f = gq + r$

$f(a) = 0 \iff (X - a) | f$

$f = a_n (X-x_1)^{\alpha_1} (X-x_2)^{\alpha_2} ...(X-x_k)^{\alpha_k}$

$x_1 + x_2 + ... + x_n = -\frac{a_{n-1}}{a_n}$

$x_1x_2...x_n = (-1)^n\frac{a_0}{a_n}$

$\|\delta\| = \max_{1\leq i\leq n} |x_i - x_{i-1}|$

$s(f; \delta) = \sum_{i=1}^n (x_i - x_{i-1})m_i$

$S(f; \delta) = \sum_{i=1}^n (x_i - x_{i-1})M_i$

$\sigma(f; \delta; \xi_i) = \sum_{i=1}^n (x_i - x_{i-1})f(\xi_i)$

$\int_a^b f(x) dx = \lim_{\|\delta_n\| \to 0} \sigma(f; \delta_n; \xi_i^n)$

$\int_a^b (\alpha f(x) + \beta g(x))dx = \alpha \int_a^b f(x)dx + \beta \int_a^b g(x)dx$

$\int_a^b f(x)dx = -\int_b^a f(x)dx$

$\int_a^b f(x)dx = F(b) - F(a)$

$\int_a^b f(x)dx = f(c) \cdot (b-a)$

$\int_a^b f(x) \cdot g'(x)dx = [f(x) \cdot g(x)]_a^b - \int_a^b f'(x) \cdot g(x)dx$

$\int_a^b (f \circ \varphi)(t) \cdot \varphi'(t)dt = \int_{\varphi(a)}^{\varphi(b)} f(x)dx$

$aria(\Gamma_f) = \int_a^b |f(x)|dx$

$vol(C_f) = \pi \int_a^b f^2(x)dx$

$l_f = \int_a^b \sqrt{1+(f'(x))^2} dx$

$D = \frac{S \cdot p \cdot n}{100}$

$D = \frac{S \cdot p \cdot m}{100 \cdot 12}$

$S_n = S_0 \cdot \left(1 + \frac{p}{100}\right)^n$

$R_p = \frac{P}{C} \cdot 100$

$R_r = \frac{P}{CT} \cdot 100$

$R_r = \frac{P}{CA} \cdot 100$

$z = a + bi$

$\overline{z} = a - bi$

$|z| = \sqrt{a^2 + b^2}$

$|z_1 + z_2| \leq |z_1| + |z_2|$

$z = \frac{z_1 + kz_2}{1 + k}$

$z = \frac{z_1 + z_2 + z_3}{3}$

$|z - z_1| = r$

$m(\angle M_3M_1M_2) = \arg \frac{z_3 - z_1}{z_2 - z_1}$

$z = r(\cos \varphi + i \sin \varphi)$

$z_1 \cdot z_2 = r_1r_2[\cos (\varphi_1 + \varphi_2) + i \sin (\varphi_1 + \varphi_2)]$

$z_1^n = r_1^n(\cos n\varphi_1 + i \sin n\varphi_1)$

$P_n = n!$

$A_n^k = \frac{n!}{(n-k)!}$

$C_n^k = \frac{n!}{k!(n-k)!}$

$C_n^k = C_n^{n-k}$

$C_n^k = C_{n-1}^k + C_{n-1}^{k-1}$

$(a+b)^n = \sum_{k=0}^n C_n^k \cdot a^{n-k} \cdot b^k$

$T_{k+1} = C_n^k \cdot a^{n-k} \cdot b^k$

$\sum_{k=0}^n C_n^k = 2^n$

$\frac{T_{k+1}}{T_k} = \frac{n-k+1}{k} \cdot \frac{b}{a}$

$\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma(1) & \sigma(2) & \cdots & \sigma(n) \end{pmatrix}$

$\text{card}(S_n) = n!$

$\varepsilon(\sigma) = (-1)^{m(\sigma)}$

$\varepsilon(\sigma_1 \circ \sigma_2) = \varepsilon(\sigma_1) \cdot \varepsilon(\sigma_2)$

$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$

$A_n^k = \frac{n!}{(n-k)!}$

$A_n^n = n!$

$A_n^k = n \cdot A_{n-1}^{k-1}$

$A_n^k = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot (n-k+1)$

$A_n^k = C_n^k \cdot k!$

$f = \frac{[C]}{[P]}$

$M = \frac{x_1 + x_2 + ... + x_n}{n}$

$A = \frac{|x_1 - M| + |x_2 - M| + ... + |x_n - M|}{n}$

$D = \sqrt{\frac{1}{n}\sum_{k=1}^n (x_k - M)^2}$

$P(A) = \frac{\text{numărul cazurilor favorabile}}{\text{numărul cazurilor posibile}} = \frac{m}{n}$

$P_B(A) = \frac{P(A \cap B)}{P(B)}$

$P(A \setminus B) = P(A) - P(A \cap B)$

$P(A) = 1 - P(\overline{A})$

$P(B) = P(A_1) \cdot P_{A_1}(B) + P(A_2) \cdot P_{A_2}(B) + ... + P(A_n) \cdot P_{A_n}(B)$

$P(X = m) = C_n^m \cdot p^m \cdot q^{n-m}$

$P(X = n) = \frac{C_a^n \cdot C_b^{k-n}}{C_{a+b}^k}$

$M(X) = \sum_{k=1}^n x_k \cdot p_k$

$D^2(X) = \sum_{k=1}^n (x_k - M(X))^2 \cdot p_k$

$(M, *)$

$(G, *)$

$H \subseteq G$

$f(x * y) = f(x) \circ f(y)$

$(A, +, \cdot)$

$a \cdot x = b$

$\mathbb{Z}_n = \{0, 1, 2, ..., n-1\}$

$\sqrt{2}$

$\pi$

$R = Q \cup (R \setminus Q)$

$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

$\sqrt{x^2} = |x|$

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

$\sqrt[n]{x^n} = x$

$\frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

$ax^2 + bx + c = 0, \quad a \neq 0, \quad a,b,c \in \mathbb{R}$

$f(x) = ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a}$

$\Delta = b^2 - 4ac$

$x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}$

$S = x_1 + x_2 = -\frac{b}{a}, \quad P = x_1 \cdot x_2 = \frac{c}{a}$

$f(x) = ax^2 + bx + c, a\neq 0$

$V\left(-\frac{b}{2a}, -\frac{\Delta}{4a}\right)$

$x = -\frac{b}{2a}$

$Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

$|a| = \begin{cases} a, & \text{dacă } a \geq 0 \\ 0, & \text{dacă } a = 0 \\ -a, & \text{dacă } a < 0 \end{cases}$

$|a \cdot b| = |a| \cdot |b|$

$\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$

$|a| - |b| \leq |a + b| \leq |a| + |b|$

$a^n = \underbrace{a \cdot a \cdot a \cdot ... \cdot a}_{\text{de n ori}}$

$a^m \cdot a^n = a^{m+n}$

$(a^m)^n = a^{m \cdot n}$

$\emptyset$

$a \in E$

$a \notin E$

$A \subset B$

$A \cup B = \{x | x \in A \text{ sau } x \in B\}$

$A \cap B = \{x | x \in A \text{ și } x \in B\}$

$A \cap B = \emptyset$

$A - B = \{x | x \in A \text{ și } x \notin B\}$

$\neg p$

$p \wedge q$

$p \vee q$

$p \rightarrow q$

$p \leftrightarrow q$

$\begin{array}{|c|c|c|c|c|c|c|} p & q & \neg p & p \wedge q & p \vee q & p \rightarrow q & p \leftrightarrow q \\ \hline 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ \hline 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ \hline 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ \hline 0 & 0 & 1 & 0 & 0 & 1 & 1 \end{array}$

$*: M \times M \to M$

$(\forall) x, y \in H \Rightarrow x * y \in H$

$(x * y) * z = x * (y * z)$

$x * y = y * x$

$x * e = e * x = x$

$x * x^{-1} = x^{-1} * x = e$

$F'(x) = f(x), \forall x \in I$

$\int f(x) dx = F(x) + C, C \in \mathbb{R}$

$\int (f + g)(x) dx = \int f(x) dx + \int g(x) dx$

$\int (f \circ \varphi)(x) \cdot \varphi'(x) dx = (F \circ \varphi)(x) + C$

$\int u(x) \cdot v'(x) dx = u(x) \cdot v(x) - \int u'(x) \cdot v(x) dx$

$\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0} = f'(x_0)$

$f'_s(x_0) = f'_d(x_0) = f'(x_0)$

$(f + g)' = f' + g'$

$(\lambda f)' = \lambda f'$

$(f \cdot g)' = f' \cdot g + f \cdot g'$

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

$(f \circ u)' = (f' \circ u) \cdot u'$

$(f^{-1})' = \frac{1}{f' \circ f^{-1}}$

$a = b \cdot c$

$a | 0$

$c.m.m.d.c.(a, b) \cdot c.m.m.m.c.(a, b) = a \cdot b$

$m_a = \frac{x + y}{2}$

$m_a = \frac{x_1 + x_2 + ... + x_n}{n}$

$m_g = \sqrt{x \cdot y}$

$m_g = \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}$

$m_h = \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2xy}{x+y}$

$m_h = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}}$

$m_p = \frac{p \cdot x + q \cdot y}{p + q}$

$m_p = \frac{p_1 \cdot x_1 + p_2 \cdot x_2 + ... + p_n \cdot x_n}{p_1 + p_2 + ... + p_n}$

$m_{pătratică} = \sqrt{\frac{a^2 + b^2}{2}}$

$\lim_{x\to x_0} f(x) = l$

$\lim_{x \to x_0^-} f(x) = l_s$

$\lim_{x \to x_0^+} f(x) = l_d$

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x \to 0} \frac{\tg x}{x} = 1$

$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

$\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

$f(x) = x^{2n}$

$g^{-1}(x) = \sqrt[2n]{x}$

$f(x) = x^{2n+1}$

$\begin{cases}mx + n = y \\ax^2 + bx + c = y\end{cases}$

$ax + b = 0$

$x = -\frac{b}{a}$

$ax + b > 0$

$x > -\frac{b}{a}$

$\begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases}$

$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$

$\det(A) = \sum_{\sigma \in S_n} \varepsilon(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}...a_{n\sigma(n)}$

$\det(A) = a_{11}a_{22} - a_{12}a_{21}$

$\det(AB) = \det(A) \cdot \det(B)$

$A \cdot A^{-1} = A^{-1} \cdot A = I_n$

$A^{-1} = \frac{1}{\det(A)} \cdot A^*$

$(u^n)' = nu^{n-1} \cdot u', n \in \mathbb{N}$

$(u^r)' = ru^{r-1} \cdot u', r \in \mathbb{R}$

$(\sqrt{u})' = \frac{1}{2\sqrt{u}} \cdot u'$

$(\frac{1}{u})' = -\frac{1}{u^2} \cdot u'$

$(a^u)' = a^u \ln a \cdot u', a \in \mathbb{R}_+, a \neq 1$

$(e^u)' = e^u \cdot u'$

$(\ln u)' = \frac{1}{u} \cdot u'$

$(\sin u)' = \cos u \cdot u'$

$(\cos u)' = -\sin u \cdot u'$

$(\tg u)' = \frac{1}{\cos^2 u} \cdot u'$

$(\ctg u)' = -\frac{1}{\sin^2 u} \cdot u'$

$(\arcsin u)' = \frac{1}{\sqrt{1-u^2}} \cdot u'$

$(\arccos u)' = -\frac{1}{\sqrt{1-u^2}} \cdot u'$

$(\arctg u)' = \frac{1}{1+u^2} \cdot u'$

$(\arcctg u)' = -\frac{1}{1+u^2} \cdot u'$

$(\sh u)' = \ch u \cdot u'$

$(\ch u)' = \sh u \cdot u'$

$\int dx = x + C$

$\int x^a dx = \frac{x^{a+1}}{a+1} + C, a \in \mathbb{R}, a \neq -1$

$\int \frac{1}{x} dx = \ln |x| + C$

$\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \arctan \frac{x}{a} + C, a \neq 0$

$\int \frac{1}{x^2 - a^2} dx = \frac{1}{2a} \ln |\frac{x-a}{x+a}| + C, a \neq 0$

$\int \frac{1}{\sqrt{x^2 ± a^2}} dx = \ln |x + \sqrt{x^2 ± a^2}| + C, a \neq 0$

$\int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin \frac{x}{a} + C, a > 0$

$\int a^x dx = \frac{a^x}{\ln a} + C, a > 0, a \neq 1$

$\int e^x dx = e^x + C$

$\int \sin x dx = -\cos x + C$

$\int \cos x dx = \sin x + C$

$\int \frac{1}{\cos^2 x} dx = \tg x + C$

$\int \frac{1}{\sin^2 x} dx = -\ctg x + C$

$\int \frac{1}{\sin x} dx = \ln |\tg \frac{x}{2}| + C$

$\int \frac{1}{\cos x} dx = \ln |\tg (\frac{x}{2} + \frac{\pi}{4})| + C$

$\int \tg x dx = -\ln |\cos x| + C$

$\int \ctg x dx = \ln |\sin x| + C$

$P(m) \land (\forall k \ge m, P(k) \rightarrow P(k + 1)) \Rightarrow \forall n \ge m, P(n)$

$k_1 \cdot k_2 \cdot ... \cdot k_n$

$n^m$

$2^n$

$n! = 1 \cdot 2 \cdot 3 \cdot ... \cdot n$

$a^n = \underbrace{a \cdot a \cdot ... \cdot a}_{n \text{ factori}}$

$c^{-n} = \frac{1}{c^n}$

$a^r = a^{\frac{m}{n}} = \sqrt[n]{a^m}$

$f : A \to B, \forall x, y \in A, x \neq y \Rightarrow f(x) \neq f(y)$

$f : A \to B, \forall y \in B, \exists x \in A : f(x) = y$

$f : A \to B \text{ este bijectivă } \Leftrightarrow f \text{ este injectivă și surjectivă}$

$f : A \to B \text{ este inversabilă } \Leftrightarrow \exists g : B \to A, f \circ g = 1_B \text{ și } g \circ f = 1_A$

$f : I \to \mathbb{R}, \forall x, y \in I, \forall a, b \geq 0, a + b = 1 : f(ax + by) \leq af(x) + bf(y)$

$f : I \to \mathbb{R}, \forall x, y \in I, \forall a, b \geq 0, a + b = 1 : f(ax + by) \geq af(x) + bf(y)$

$a : b$ sau $b | a$

$D_a = \{x | x | a, x \in \mathbb{N}\}$

$M_a = \{a \cdot n | n \in \mathbb{N}\}$

$(a, b) = d$ sau $\text{c.m.m.d.c.}(a, b) = d$

$[a, b] = m$ sau $\text{c.m.m.m.c.}(a, b) = m$

$2 | n$

$3 | (d_1 + d_2 + ... + d_k)$

$4 | (10a + b)$

$5 | n$ sau $10 | (n - 5)$

$2 | n$ și $3 | n$

$8 | (100a + 10b + c)$

$9 | (d_1 + d_2 + ... + d_k)$

$11 | (d_1 - d_2 + d_3 - d_4 + ...)$

$25 | (100a + b)$

$125 | (1000a + 100b + 10c + d)$

$10^k | n$

$7 | (A - B)$ sau $11 | (A - B)$ sau $13 | (A - B)$

$\vec{u} \cdot \vec{v} = x_ux_v + y_uy_v$

$\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\sphericalangle(\vec{u},\vec{v}))$

$\vec{u} \perp \vec{v} \Leftrightarrow x_ux_v + y_uy_v = 0$

$\vec{u} \parallel \vec{v} \Leftrightarrow \frac{x_u}{x_v} = \frac{y_u}{y_v}$

$a^2 = b^2 + c^2 - 2bc \cos A$

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

$\sin \frac{A}{2} = \sqrt{\frac{(p-b)(p-c)}{bc}}$

$\cos \frac{A}{2} = \sqrt{\frac{p(p-a)}{bc}}$

$S = \frac{ab \sin C}{2}$

$S = \sqrt{p(p-a)(p-b)(p-c)}$

$m_a^2 = \frac{2(b^2 + c^2) - a^2}{4}$

$l_a = \frac{2\sqrt{bc\cdot p(p-a)}}{b+c}$

$a^2 \cdot m + c^2 \cdot n = b^2 \cdot (m+n) + mn \cdot (m+n)$

$3x^2y, -5ab^3, 2xyz$

$3x^2y^3z$

$grad(3x^2y^3z) = 2 + 3 + 1 = 6$

$(3x^2y) \cdot (2xy^3) = 6x^3y^4$

$(3x^2y)^3 = 27x^6y^3$

$\frac{6x^3y^2}{2xy} = 3x^2y$

$f: A \rightarrow B$

$f: A \rightarrow B, A = \{1, 2, 3\}, B = \{a, b, c\}, f(1) = a, f(2) = b, f(3) = c$

$f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = 2x + 1$

$\{(x, y) | x \in A, y = f(x)\}$

$f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = ax + b$

$G_f \cap Oy = A(0, b)$

$G_f \cap Ox = B(-\frac{b}{a}, 0)$

$\begin{array}{|c|c|c|c|c|c|}\hline x & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ \\\hline\sin x & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1 \\\hline\end{array}$

$\begin{array}{|c|c|c|c|c|c|}\hline x & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ \\\hline\cos x & 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & 0 \\\hline\end{array}$

$\begin{array}{|c|c|c|c|c|c|}\hline x & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ \\\hline\tg x & 0 & \frac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text{nedefinit} \\\hline\end{array}$

$\sin \left(\frac{\pi}{2} - x\right) = \cos x, \quad \forall x \in \mathbb{R}$

$\cos \left(\frac{\pi}{2} - x\right) = \sin x, \quad \forall x \in \mathbb{R}$

$\sin(-x) = -\sin x, \quad \forall x \in \mathbb{R}$

$\cos(-x) = \cos x, \quad \forall x \in \mathbb{R}$

$\sin(x + 2k\pi) = \sin x, \quad \forall k \in \mathbb{Z}$

$\cos(x + 2k\pi) = \cos x, \quad \forall k \in \mathbb{Z}$

$\tg(x + k\pi) = \tg x, \quad \forall k \in \mathbb{Z}$

$\ctg(x + k\pi) = \ctg x, \quad \forall k \in \mathbb{Z}$

$f: \mathbb{R} \to (0,+\infty), f(x)=a^x$

$f(0) = a^0 = 1$

$g: (0,+\infty) \to \mathbb{R}, g(x)=\log_a x$

$g(1) = \log_a 1 = 0$

$f \circ g = g \circ f = 1_{(0,+\infty)}$