### 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\$