Posts for Tag: HSC

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

Complex Numbers – HSC Math

A complex number is a number that can be expressed in the form $a + ib$, where a and b are real numbers, and $i$ is a solution of the equation $x^2 = −1$. Because no real number satisfies this equation, $i$ is called an imaginary number. Cartesian form $z=x+iy; x=r\ cos\ θ, t=r\ sin\Read More

Real Numbers – HSC Math

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. Real number set can be represented by the symbol R. If $a>b$ and $(x-a)(x-b)b$. So $b < x < a$. If $a>b$ and $(x-a)(x-b)>0$ then $x>a$ or $x<b$. So $x < b$ or $x >Read More

Integration Formulas – HSC Math

$∫1/\x\ \dx=ln|x|+c$ $∫e^{mx}\ \dx=1/m\ e^{mx}+c$ $∫a^xd^x\ \dx=a^x/{ln\ a}+c, a>0, a≠1$ $∫cos\ x\ \dx=sin\ x+c$ $∫sin\ x\ \dx=-cos\ x+c$ $∫sec^2\ x\ \dx=tan\ x+c$ $∫cosec^2\ x\ \dx=-cot\ x+c$ $∫sec\ x\ tan\ x\ \dx=sec\ x+c$ $∫cosec\ x\ cot\ x\ \dx=-cosec\ x+c$ $∫tan\ x\ \dx=-ln|cos\ x|+c=ln|sec\ x|+c$ $∫cot\ x\ \dx=ln|sin\ x|+c$ $∫sec\ x\ \dx=ln|tan(π/4+x/2)|+c$ $∫sec\ x\ \dx=ln|sec\ x+tan\ x|+c$Read More

Application of Differentiation – HSC Math

if $(x,y) is a point over line $y=f(x)$ and the slope of the line at that point $(x,y)$ is θ$, then- $tan\ θ=\d/\dy(y)$ here, the tangent of the line at point $(x,y)$ creates θ° angle with $x\ axis$. If the tangent of a line is parallel to $x\ axis$ or perpendicular to $y\ axis$ then-Read More

Multiple Times Differentiation – HSC Math

if $y=f(x),$ then $\d/{\dx}(y)=\dy/{\dx}=f'(x)=y′=y_1$ $\d/{\dx}(y_1)=\d^2y/{\dx^2}=y_2$ $\d/{\dx}(y_2)=\d^3y/{\dx^3}=y_3$ $\d/{\dx}(y_{n-1})=\d^ny/{\dx^n}=y_n$ if $y=e^{mx}, y_n=m^n\ e^{mx}$ if $y=a^x, y_n=a^x\ (log_e\ a)^n=a^x\ (ln\ a)^n$ if $y=sin(ax+b), y_n=a^n\ sin(n.π/2+(ax+b))$ if $y=cox(ax+b), y_n=a^n\ cos(n.π/2+(ax+b))$