### All Math Formula

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### Trigonometric Equations – HSC Math

Whole number: n if \$sin\ θ=sin\ a,\$ then \$θ=nπ+(-1)^n\ a\$ if \$cosec\ θ=cosec\ a,\$ then \$θ=nπ+(-1)^n\ a\$ if \$cos\ θ=cos\ a,\$ then \$θ=2nπ±a\$ if \$sec\ θ=sec\ a,\$ then \$θ=2nπ±a\$ if \$tan\ θ=tan\ a,\$ then \$θ=nπ+a\$ if \$cot\ θ=cot\ a,\$ then \$θ=nπ+a\$ if \$sin\ θ=0,\$ then \$θ=nπ\$ if \$sin\ θ=1,\$ then \$θ=(4n+1)π/2\$ if \$sin\ θ=-1,\$ thenRead More

### Inverse Trigonometric Functions – HSC Math

\$sin^{-1}x+cos^{-1}x=π/2\$ \$tan^{-1}x+cot^{-1}x=π/2\$ \$sec^{-1}x+cosec^{-1}x=π/2\$ \$sin^{-1}(-x)=-sin^{-1}x\$ \$cos^{-1}(-x)=π-cos^{-1}x\$ \$tan^{-1}(-x)=tan^{-1}x\$ \$cot^{-1}(-x)=π-cot^{-1}x\$ \$sec^{-1}(-x)=π-sec^{-1}x\$ \$cosec^{-1}(-x)=-cosec^{-1}x\$ \$sin^{-1}(1/x)=cosec^{-1}x\$ \$cos^{-1}(1/x)=sec^{-1}x\$ \$tan^{-1}(1/x)=cot^{-1}x\$ \$sin^{-1}x+sin^{-1}y=sin^{-1}(x√{1-y^2}+y√{1-x^2})\$ \$sin^{-1}x-sin^{-1}y=sin^{-1}(x√{1-y^2}-y√{1-x^2})\$ \$cos^{-1}x+cos^{-1}y=cos^{-1}(xy-√{1-x^2}√{1-y^2})\$ \$cos^{-1}x-cos^{-1}y=cos^{-1}(xy+√{1-x^2}√{1-y^2})\$ \$tan^{-1}x+tan^{-1}y=tan^{-1}{x+y}/{1-xy}\$ \$tan^{-1}x-tan^{-1}y=tan^{-1}{x-y}/{1+xy}\$ \$2sin^{-1}(x)=sin^{-1}(2x√{1-x^2})\$ \$2cos^{-1}(x)=cos^{-1}(2x^2-1)\$ \$2tan^{-1}(x)=tan^{-1}{2x}/{1-x^2}=cos^{-1}{1-x^2}/{1+x^2}\$ \$3sin^{-1}(x)=sin^{-1}(3x-4x^3)\$ \$3cos^{-1}(x)=cos^{-1}(4x^3-3x)\$ \$3tan^{-1}(x)=tan^{-1}{3x-x^3}/{1-3x^2}\$ \$1/2sin^{-1}(x)=tan^{-1}{1-√{1-x^2}}/x\$ \$1/2cos^{-1}(x)=cos^{-1}√{{1+x}/2}=sin^{-1}√{{1-x}/2}=tan^{-1}√{{1-x}/{1+x}}\$ \$1/2tan^{-1}(x)=tan^{-1}{√{1+x^2}-1}/x\$ \$sin^{-1}(x)=cos^{-1}√{1-x^2}\$ \$=tan^{-1}x/√{1-x^2}\$ \$=cot^{-1}√{1-x^2}/x\$ \$=sec^{-1}1/√{1-x^2}\$ \$=cosec^{-1}1/x\$

### Conics Formulas – HSC Math

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse. Identification of conics from general formula: \$ax^2+by^2+2gx+2fy+2hxy+c=0\$ \$i. ab-h^2=0, Parabola\$ \$ii. ab-h^2>0, Ellipse\$ \$ii. ab-h^2<0, Hyperbola\$ \$iv.Read More

### Binomial Theorem – HSC Math

In elementary algebra, the binomial theorem or binomial expansion describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial \$(x+y)^n\$ into a sum involving terms of the form a \$ax^by^c\$, where the exponents \$b\$ and \$c\$ are nonnegative integers with \$b+c=n\$, and the coefficient \$a\$ ofRead More

### Polynomial Equation – HSC Math

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate \$x\$ is \$x^2−4x+7\$. An example in three variables is \$x^3+2xyz^2−yz+1\$. Nature of roots of a quadraticRead More

### Reduction Formulas

\$∫x^ne^{mx}\ dx=1/mx^ne^{mx}-n/m∫x^{n-1}e^{mx}\ dx\$ \$∫{e^{mx}}/{x^n}\ dx=-e^{mx}/{(n-1)x^{n-1}}+m/{n-1}∫e^{mx}/{x^{n-1}}\ dx, n≠1\$ \$∫sin\ h^nx\ dx=1/nsin\ h^{n-1}x\ cos\ hx-{n-1}/n∫sin\ h^{n-2}x\ dx\$ \$∫{dx}/{sin\ h^nx}=-{cos\ hx}/{(n-1)sin\ h^{n-1}x}-{n-2}/{n-1}∫{dx}/{sin\ h^{n-2}x},n≠1\$ \$∫cos\ h^nx\ dx=1/nsin\ hx\ cos\ h^{n-1}x\ cos\ hx+{n-1}/n∫cos\ h^{n-2}x\ dx\$ \$∫{dx}/{cos\ h^nx}=-{sin\ hx}/{(n-1)cos\ h^{n-1}x}+{n-2}/{n-1}∫{dx}/{cos\ h^{n-2}x},n≠1\$ \$∫sin\ h^nx\ cos\ h^mx\ dx={sin\ h^{n+1}x\ cos\ h^{m-1}x}/{n+m}\$ \$+{m-1}/{n+m}∫sin\ h^n\ cos\ h^{m-2}x\ dx\$ \$∫sin\ h^nx\ cos\ h^mx\ dx={sin\ h^{n-1}x\ cos\Read More

### Integral of Exponential and Logarithmic Functions

\$∫e^x\ dx=e^x+c\$ \$∫a^x\ dx={a^x}/{ln\ a}+c\$ \$∫e^{ax}\ dx=e^{ax}/a+c\$ \$∫xe^{ax}\ dx=e^{ax}/a^2(ax-1)+c\$ \$∫ln\ x\ dx=x\ ln\ x-x+c\$ \$∫{dx}/{x\ ln\ x}=ln|ln\ x|+c\$ \$∫x^n\ ln\ x\ dx=x^{n+1}[{ln\ x}/{n+1}-1/(n+1)^2]+c\$ \$∫e^{ax}\ sin\ bx\ dx={a\ sin\ bx-b\ cos\ bx}/{a^2+b^2}e^{ax}+c\$ \$∫e^{ax}\ cos\ bx\ dx={a\ cos\ bx+b\ sin\ bx}/{a^2+b^2}e^{ax}+c\$

### Integrals of Hyperbolic Functions

\$∫sin\ hx\ dx=cos\ hx+c\$ \$∫cos\ hx\ dx=sin\ hx+c\$ \$∫tan\ hx\ dx=ln\ cos\ hx+c\$ \$∫cot\ hx\ dx=ln|sin\ hx|+c\$ \$∫sec\ h^2x\ dx=tan\ hx+c\$ \$∫cosec\ h^2x\ dx=-cot\ hx+c\$ \$∫sec\ hx\ tan\ hx\ dx=-sec\ hx+c\$ \$∫cosec\ hx\ cot\ hx\ dx=-cosec\ hx+c\$

### Integrals of Trigonometric Functions

\$∫sin\ x\ dx=-cos\ x+c\$ \$∫cos\ x\ dx=sin\ x+c\$ \$∫sin^2\ x \ dx=x/2-1/4sin\ 2x+c\$ \$∫cos^2\ x \ dx=x/2+1/4sin\ 2x+c\$ \$∫sin^3\ x \ dx=1/3cos^3\ x-cos\ x+c=1/12cos\ 3x-3/4cos\ x+c\$ \$∫cos^3\ x \ dx=sin\ x-1/3sin^3\ x+c=1/12sin\ 3x+3/4sin\ x+c\$ \$∫{dx}/{sin\ x}=∫cosec\ x\ dx=ln|tan{x/2}|+c\$ \$∫{dx}/{cos\ x}=∫sec\ x\ dx=ln|tan(π/4+x/2)|+c\$ \$∫{dx}/{sin^2\ x}=∫cosec^2\ x\ dx=-cot\ x+c\$ \$∫{dx}/{cos^2\ x}=∫sec^2\ x\ dx=tan\ x+c\$ \$∫{dx}/{sin^3\ x}=∫cosec^3\ x\Read More