Algebra & Trigonometry Essentials

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1. Polynomial Equations General Form: $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0$ Imaginary Roots: If coefficients are real, imaginary roots occur in conjugate pairs ($a+bi, a-bi$). Irrational Roots: If coefficients are rational, irrational roots occur in conjugate pairs ($a+\sqrt{b}, a-\sqrt{b}$). Relation between Roots and Coefficients (Vieta's Formulas): For $x^n + p_1 x^{n-1} + \dots + p_n = 0$ with roots $\alpha_1, \dots, \alpha_n$: $\sum \alpha_i = -p_1$ $\sum_{i $\dots$ $\alpha_1 \alpha_2 \dots \alpha_n = (-1)^n p_n$ Symmetric Functions of Roots: Expressible in terms of elementary symmetric polynomials (coefficients). Transformation of Equation: Roots are $k$ times the original: replace $x$ with $x/k$. Roots are $k$ less than original: replace $x$ with $x+k$. Reciprocal Equation: $a_n x^n + \dots + a_1 x + a_0 = 0$ where $a_k = a_{n-k}$. If $\alpha$ is a root, $1/\alpha$ is also a root. Removal of Terms: To remove the second term of $x^n + a_1 x^{n-1} + \dots = 0$, substitute $x = y - a_1/n$. Descartes's Rule of Signs: Number of positive real roots $\le$ number of sign changes in $P(x)$. Number of negative real roots $\le$ number of sign changes in $P(-x)$. Cardan's Method (Cubic $x^3+px+q=0$): Let $x=u+v$. $u^3+v^3 = -q$, $3uv=-p$. Solve for $u^3, v^3$ using quadratic. 2. Matrix Algebra Symmetric Matrix: $A^T = A$ ($a_{ij} = a_{ji}$) Skew-Symmetric Matrix: $A^T = -A$ ($a_{ij} = -a_{ji}$, diagonal elements are 0) Hermitian Matrix: $A^* = A$ ($a_{ij} = \overline{a_{ji}}$) Skew-Hermitian Matrix: $A^* = -A$ ($a_{ij} = -\overline{a_{ji}}$) Orthogonal Matrix: $A^T A = A A^T = I$ (implies $A^{-1} = A^T$, $\det(A) = \pm 1$) Unitary Matrix: $A^* A = A A^* = I$ (implies $A^{-1} = A^*$, $|\det(A)| = 1$) Eigenvalues ($\lambda$) and Eigenvectors ($\mathbf{v}$): $A\mathbf{v} = \lambda\mathbf{v}$, where $\mathbf{v} \ne \mathbf{0}$. Characteristic Equation: $\det(A - \lambda I) = 0$. Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation. If $P(\lambda) = \det(A - \lambda I)$, then $P(A) = \mathbf{0}$. Similar Matrices: $A$ and $B$ are similar if $B = P^{-1} A P$ for some invertible matrix $P$. Similar matrices have the same eigenvalues, determinant, trace. Diagonalization: A matrix $A$ is diagonalizable if it is similar to a diagonal matrix $D$. $D = P^{-1} A P$, where columns of $P$ are eigenvectors of $A$, and diagonal elements of $D$ are corresponding eigenvalues. 3. Number Theory Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself. Composite Number: A natural number greater than 1 that is not prime. Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely represented as a product of prime numbers (up to the order of the factors). Divisor of an Integer: An integer $d$ is a divisor of $n$ if $n = kd$ for some integer $k$. Euler's Totient Function ($\phi(n)$): Counts the number of positive integers up to a given integer $n$ that are relatively prime to $n$. If $n = p_1^{k_1} \dots p_r^{k_r}$, then $\phi(n) = n \prod_{i=1}^r (1 - 1/p_i)$. Congruence Modulo $n$: $a \equiv b \pmod n$ if $n$ divides $(a-b)$. Highest Power of Prime $p$ in $n!$ (Legendre's Formula): $E_p(n!) = \sum_{k=1}^\infty \left\lfloor \frac{n}{p^k} \right\rfloor$. Fermat's Little Theorem: If $p$ is a prime number, then for any integer $a$ not divisible by $p$, $a^{p-1} \equiv 1 \pmod p$. Also, $a^p \equiv a \pmod p$ for all $a$. Wilson's Theorem: An integer $p > 1$ is a prime number if and only if $(p-1)! \equiv -1 \pmod p$. Lagrange's Theorem (Polynomial Congruences): If $P(x)$ is a polynomial of degree $n$ with integer coefficients, and $p$ is a prime, then the congruence $P(x) \equiv 0 \pmod p$ has at most $n$ solutions modulo $p$, unless all coefficients of $P(x)$ are multiples of $p$. 4. Summation of Series Binomial Series: $(1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k$ for integer $n \ge 0$. For any real $n$ and $|x| Exponential Series: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ for all $x \in \mathbb{R}$. Logarithmic Series: $\ln(1+x) = \sum_{n=1}^\infty (-1)^{n-1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ for $-1 $\ln(1-x) = -\sum_{n=1}^\infty \frac{x^n}{n} = -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots$ for $-1 \le x C + i S method: For sums involving sines and cosines, form a complex sum $C+iS$ using $e^{i\theta} = \cos\theta + i\sin\theta$. Sum the geometric series and take real/imaginary parts. Telescopic Summation: $\sum_{k=1}^n (a_k - a_{k+1}) = a_1 - a_{n+1}$. 5. Trigonometry De Moivre's Theorem: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$. Expansion of $\sin(nX)$, $\cos(nX)$, $\tan(nX)$: Use De Moivre's theorem for $\cos(nX) + i\sin(nX) = (\cos X + i\sin X)^n$. For $\tan(nX)$, use $\tan(nX) = \frac{\sin(nX)}{\cos(nX)}$. Expansion of $\sin^n X$, $\cos^n X$: Use $\cos X = \frac{e^{iX} + e^{-iX}}{2}$ and $\sin X = \frac{e^{iX} - e^{-iX}}{2i}$. Taylor Series Expansions: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$ $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}$ $\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots$ (for $|x| Sum of Roots of Trigonometric Equations: For equations like $\sin x = k$, roots are $n\pi + (-1)^n \alpha$. Sum depends on the interval. Formation of Equations with Trigonometric Roots: Use Vieta's formulas. E.g., if roots are $\cos(2\pi/n), \cos(4\pi/n), \dots$. 6. Hyperbolic Functions Definitions: $\sinh x = \frac{e^x - e^{-x}}{2}$ $\cosh x = \frac{e^x + e^{-x}}{2}$ $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ Identities: $\cosh^2 x - \sinh^2 x = 1$ $\text{sech}^2 x + \tanh^2 x = 1$ $\coth^2 x - \text{csch}^2 x = 1$ $\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y$ $\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y$ Relation to Circular Functions: $\sin(ix) = i\sinh x$ $\cos(ix) = \cosh x$ $\tan(ix) = i\tanh x$ $\sinh(ix) = i\sin x$ $\cosh(ix) = \cos x$ $\tanh(ix) = i\tan x$ Inverse Hyperbolic Functions: (Logarithmic forms) $\text{arsinh } x = \ln(x + \sqrt{x^2+1})$ $\text{arccosh } x = \ln(x + \sqrt{x^2-1})$, for $x \ge 1$ $\text{artanh } x = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)$, for $|x| 7. Complex Numbers Logarithm of a Complex Number: For $z = x+iy = re^{i\theta}$: $\ln z = \ln(re^{i(\theta + 2k\pi)}) = \ln r + i(\theta + 2k\pi)$, where $k \in \mathbb{Z}$. Principal Value: $\text{Ln } z = \ln r + i\theta$, where $-\pi