# 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))\$