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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