Trigonometric Properties of a Triangle – HSC Math

Sine Rule $a/{Sin\ A}=b/{Sin\ B}=c/{Sin\ C}$ Cos Rule $cos\ A={b^2+c^2-a^2}/{2bc}$ $cos\ B={a^2+c^2-b^2}/{2ac}$ $cos\ C={a^2+b^2-c^2}/{2ab}$ $a=b\ cos\ C+c\ cos\ B$ $b=c\ cos\ A+a\ cos\ C$ $c=a\ cos\ B+b\ cos\ A$ Tan Rule $tan\ {B-C}/2={b-c}/{b+c}\ cot\ A/2$ $tan\ {A-B}/2={a-b}/{a+b}\ cot\ C/2$ $tan\ {C-A}/2={c-a}/{c+a}\ cot\ B/2$ $tan\ A/2=√{(s-b)(s-c)}/√{s(s-a)}={(s-b)(s-c)}/{∆}$ $tan\ B/2=√{(s-a)(s-c)}/√{s(s-c)}={(s-a)(s-c)}/{∆}$ $tan\ C/2=√{(s-a)(s-b)}/√{s(s-c)}={(s-a)(s-b)}/{∆}$ Here, $s={a+b+c}/2, ∆=$Area of that triangle.Read More

Trigonometric Formulas – HSC Math

In circular system, $θ°={\arc}/{\radius}=s/r$ or, $s=rθ$ $1 \radian=1 \rad=1^c=(180/π)^°$ Area of a circular sector (symbol ⌔) $⌔\AOB=1/2r^2θ$ [θ in radian] $sin\ θ=1/{cosec\ θ}; cos\ θ=1/{sec\ θ}; tan\ θ=1/{cot\ θ}$ $tan\ θ={sin \θ}/{cos\ θ}; cot\ θ={cos\ θ}/{sin\ θ}$ $sin^2\ θ+cos^2\ θ=1$ $sec^2\ θ-tan^2\ θ=1$ $cosec^2\ θ-cot^2\ θ=1$ $-1≤sin\ θ≤1$ $-1≤cos\ θ≤1$ $cosec\ θ≤-1, 1≤cosec\ θ$ $sec\Read More

Operations with matrices

Two matroces A and B are equal if, and only if, they are both of the same shape $mxn$ and corresponding elemts are equal, i.e. $[a_{ij}\ ]=[b_{ij}\ ]$ Two matrices A and B can be added (or subtracted) if, and only if, they have the same shape $mxn$. And in the resultant matrix each ofRead More

Matrices Formulas

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. An $m\ x\ n$ matrix A is a rectangular array of elements (numbers of functions) with m rows and n columns $A=[a_{ij}]=[\table a_11,a_12,…,a_{1n};a_21,a_22,…,a_{2n};…,…,…,…;a_{m1},a_{m2},…,a_{mn}]$ Square matrix is a matrix of order $nxn$ A square matrix $[a_{ij}]$Read More

Properties of Determinants

The value of determinant remains unchanged if rows are changed to columns and columns are changed to rows. $|\table a_1, a_2; b_1, b_2|=|\table a_1, b_1; a_1, b_2|$ If two rows or two columns are interchanged, the sign of the determinant is changed. $|\table a_1, a_2; b_1, b_2|=-|\table b_1, b_2; a_1, a_2|$ If two rows orRead More

Determinants Formulas

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Matrices: A Elements of a matrix: ai, ai, aij, bij, … aij… Determinant of matrix A: |A| Second order determinant $|A|=|\table a_1,b_1;a_2,b_2|=a_1b_2-a_2b_1$ ThirdRead More

Relations to Hyperbolic Functions

Imaginary unit: i $sin(ix)=i\ sinhx$ $tan(ix)=i\ tanhx$ $cot(ix)=-i\ cothx$ $sec(ix)=sechx$ $cosec(ix)=-i\ cosechx$

Trigonometric Equations

Whole number: n if $sin\ x=a,$ then $x=πn+(-1)^n sin^{-1}\ a$ if $sin\ x=sin\ a,$ then $x=πn+(-1)^n\ a$ if $cos\ x=a,$ then $x=2πn±cos^{-1}\ a$ if $cos\ x=cos\ a,$ then $x=2πn±a$ if $tan\ x=a,$ then $x=πn+tan^{-1}\ a$ if $tan\ x=tan\ a,$ then $x=πn+a$ if $cot\ x=a,$ then $x=πn+cot^{-1}\ a$ if $cot\ x=cot\ a,$ then $x=πn+a$ if $sec\Read More

Circular Torus Formulas

Major radius: R Minor radius: r Surface area: S Volume: V $S=4π^2Rr$ $V=2π^2Rr^2$

Oblate Spheroid Formulas

An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis. Semi-Axes: a, b, b (a<b) Surface Area: S Volume: V $S=2πb(b+{a\ sin^{-1}e)}/e,$ where $e=√{a^2-b^2}/a$ $V=4/3πb^2a$