Math Formula Book

\$A = s^2\$

\$P = 4s\$

\$A = l * b\$

\$P = 2(l + b)\$

\$A = pi * r^2\$

\$P = 2pi * r\$

\$A = pi * r * theta\$

Where \$theta\$ is the angle of sector, in radians. \$r\$ is the radius of circle.

\$A = 1/2 * b * h\$

\$s = {a + b + c} / 2\$

\$A = sqrt {s * (s - a) * (s - b) * (s - c)}\$

\$P = a + b + c\$

\$A = b * h\$

Complex Number Standard Form:

\$z = a + bi\$

where a is the real part, b is the imaginary part, and i is the imaginary unit.

Imaginary Unit:

\$i^2 = -1\$

\$i^3 = -i\$

\$i^4 = 1\$

\$i^(4k) = 1, i^(4k+1) = i, i^(4k+2) = -1, i^(4k+3) = -i\$

Real and Imaginary Parts:

\$Re(z) = a\$

\$Im(z) = b\$

Definition:

\$bar z = conj(z) = a - bi\$

Properties:

\$z + bar z = 2a = 2*Re(z)\$

\$z - bar z = 2bi = 2i*Im(z)\$

\$z * bar z = a^2 + b^2 = |z|^2\$

\$bar (bar z) = z\$

\$bar (z1 + z2) = bar z1 + barz2\$

\$bar (z1 * z2) = bar z1 * bar z2\$

\$bar frac {z1} {z2} = bar (z1) / bar (z2)\$

Definition:

\$|z| = sqrt(a^2 + b^2) = sqrt(z * z*)\$

Properties:

\$|z| >= 0\$

\$|z| = 0 " if and only if " z = 0\$

\$|z1 * z2| = |z1| * |z2|\$

\$|z1/z2| = |z1|/|z2| " " (z2 != 0)\$

\$|z1 + z2| <= |z1| + |z2|\$ (Triangle Inequality)

\$||z1| - |z2|| <= |z1 - z2|\$

\$|z^n| = |z|^n\$

Definition:

\$arg(z) = theta = arctan(b/a)\$ (with appropriate quadrant adjustment)

Principal Argument:

\$Arg(z) = theta " where " -pi < theta <= pi\$

Properties:

\$arg(z1 * z2) = arg(z1) + arg(z2) + 2pik\$

\$arg(z1/z2) = arg(z1) - arg(z2) + 2pik\$

\$arg(z^n) = n * arg(z) + 2pik\$

\$arg(z*) = -arg(z) + 2pik\$

Polar Representation:

\$z = r * (cos(theta) + i*sin(theta)) = r * e^(itheta)\$

where \$r = |z|\$ and \$theta = arg(z)\$

Euler’s Formula:

\$e^(itheta) = cos(theta) + i*sin(theta)\$ \$e^(-itheta) = cos(theta) - i*sin(theta)\$

Conversion Formulas:

Cartesian to Polar: \$r = sqrt(a^2 + b^2)\$ \$theta = arctan(b/a)\$ (with quadrant correction)

Polar to Cartesian: \$a = r * cos(theta)\$ \$b = r * sin(theta)\$

Addition:

\$(a1 + b1*i) + (a2 + b2*i) = (a1 + a2) + (b1 + b2)*i\$

Subtraction:

\$(a1 + b1*i) - (a2 + b2*i) = (a1 - a2) + (b1 - b2)*i\$

Multiplication:

\$(a1 + b1*i) * (a2 + b2*i) = (a1*a2 - b1*b2) + (a1*b2 + b1*a2)*i\$

Division:

\$(a1 + b1*i) / (a2 + b2*i) = [(a1*a2 + b1*b2) + (b1*a2 - a1*b2)*i\$ / (a2^2 + b2^2)]

Polar Form Operations:

\$z1 * z2 = r1*r2 * e^(i(theta1 + theta2))\$ \$z1 / z2 = (r1/r2) * e^(i(theta1 - theta2))\$

De Moivre’s Theorem:

\$z^n = r^n * e^(i*n*theta) = r^n * (cos(n*theta) + i*sin(n*theta))\$

nth Roots:

\$z^(1/n) = r^(1/n) * e^(i*(theta + 2pik)/n)\$

where k = 0, 1, 2, …​, n-1 gives all n distinct roots.

Square Roots:

\$sqrt(a + bi) = +-[sqrt((r + a)/2) + i*sgn(b)*sqrt((r - a)/2)\$]

where \$r = |a + bi|\$ and sgn(b) is the sign of b.

Principal nth Root:

\$z^(1/n) = |z|^(1/n) * e^(i*Arg(z)/n)\$

Complex Exponential:

\$e^z = e^(a+bi) = e^a * e^(bi) = e^a * (cos(b) + i*sin(b))\$

Properties of Complex Exponential:

\$e^(z1 + z2) = e^z1 * e^z2\$ \$e^(z1 - z2) = e^z1 / e^z2\$ \$(e^z)^n = e^(n*z)\$ \$|e^z| = e^(Re(z))\$ \$arg(e^z) = Im(z) + 2pik\$

Complex Logarithm:

\$log(z) = ln(|z|) + i*(arg(z) + 2pik)\$

Principal Logarithm:

\$Log(z) = ln(|z|) + i*Arg(z)\$

Properties:

\$log(z1 * z2) = log(z1) + log(z2) + 2piki\$

\$log(z1/z2) = log(z1) - log(z2) + 2piki\$

\$log(z^n) = n*log(z) + 2piki\$

\$e^(log(z)) = z\$

\$log(e^z) = z + 2piki\$

Complex Sine:

\$sin(z) = (e^(iz) - e^(-iz)) / (2i)\$

\$sin(x + iy) = sin(x)*cosh(y) + i*cos(x)*sinh(y)\$

Complex Cosine:

\$cos(z) = (e^(iz) + e^(-iz)) / 2\$

\$cos(x + iy) = cos(x)*cosh(y) - i*sin(x)*sinh(y)\$

Complex Tangent:

\$tan(z) = sin(z) / cos(z) = (e^(iz) - e^(-iz)) / (i*(e^(iz) + e^(-iz)))\$

\$tan(x + iy) = [sin(2x) + i*sinh(2y)\$ / [cos(2x) + cosh(2y)]]

Fundamental Identities:

\$sin^2(z) + cos^2(z) = 1\$

\$sin(z + 2pi) = sin(z)\$

\$cos(z + 2pi) = cos(z)\$

\$tan(z + pi) = tan(z)\$

Complex Hyperbolic Sine:

\$sinh(z) = (e^z - e^(-z)) / 2\$

\$sinh(x + iy) = sinh(x)*cos(y) + i*cosh(x)*sin(y)\$

Complex Hyperbolic Cosine:

\$cosh(z) = (e^z + e^(-z)) / 2\$

\$cosh(x + iy) = cosh(x)*cos(y) + i*sinh(x)*sin(y)\$

Complex Hyperbolic Tangent:

\$tanh(z) = sinh(z) / cosh(z)\$

\$tanh(x + iy) = [sinh(2x) + i*sin(2y)\$ / [cosh(2x) + cos(2y)]]

Relationships with Trigonometric Functions:

\$sin(iz) = i*sinh(z)\$

\$cos(iz) = cosh(z)\$

\$sinh(iz) = i*sin(z)\$

\$cosh(iz) = cos(z)\$

Common Values:

\$e^(ipi) = -1\$ (Euler’s Identity)

\$e^(ipi/2) = i\$

\$e^(ipi/4) = (1 + i)/sqrt(2)\$

\$e^(2pii) = 1\$

Useful Identities:

\$cos(theta) = (e^(itheta) + e^(-itheta)) / 2\$

\$sin(theta) = (e^(itheta) - e^(-itheta)) / (2i)\$

\$1 + e^(itheta) = 2*cos(theta/2) * e^(itheta/2)\$

\$1 - e^(itheta) = -2i*sin(theta/2) * e^(itheta/2)\$

Distance Formula:

\$|z1 - z2| = " distance between " z1 " and " z2 " in complex plane"\$

Multiplication by i:

\$i * z " rotates " z " by " 90° " counterclockwise"\$

Multiplication by e^(iθ):

\$e^(itheta) * z " rotates " z " by angle " theta\$

Reflection:

\$z* " reflects " z " across the real axis"\$

Exponential Series:

\$e^z = sum_(n=0)^oo (z^n)/(n!)\$

Sine Series:

\$sin(z) = sum_(n=0)^oo ((-1)^n * z^(2n+1))/((2n+1)!)\$

Cosine Series:

\$cos(z) = sum_(n=0)^oo ((-1)^n * z^(2n))/((2n)!)\$

Geometric Series:

\$1/(1-z) = sum_(n=0)^oo z^n " for " |z| < 1\$

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\$I = {PNR} / 100\$

\$A = P (1 + R / 100)^N\$

\$I = A - P = P (1 + R / 100)^N - P\$

\$I = P (1 + R / 100)^N - P\$

\$E = P * R * (1 + R)^N / ((1 + R)^N - 1)\$

\$hyp^2 = sqrt {opp^2 + adj^2}\$

\$sin(theta) = {opp} / {hyp}\$

\$cos(theta) = {adj} / {hyp}\$

\$tan(theta) = {opp} / {adj}\$

\$cot(theta) = {hyp} / {opp}\$

\$sec(theta) = {adj} / {hyp}\$

\$cosec(theta) = {hyp} / {adj}\$

\$cosec(theta) = 1 / {sin(theta)}\$

\$sec(theta) = 1 / {cos(theta)}\$

\$cot(theta) = 1 / {tan(theta)}\$

\$sin(theta) = 1 / {cosec(theta)}\$

\$cos(theta) = 1 / {sec(theta)}\$

\$tan(theta) = 1 / {cot(theta)}\$

\$tan(theta) = {sin(theta)} / {cos(theta)}\$

\$tan(theta) = {sec(theta)} / {cosec(theta)}\$

\$cot(theta) = {cos(theta)} / {sin(theta)}\$

\$cot(theta) = {cosec(theta)} / {sec(theta)}\$

\$sin^2(theta) + cos^2(theta) = 1\$

\$sec^2(theta) - tan^2(theta) = 1\$

\$cosec^2(theta) - cot^2(theta) = 1\$

\$sin(90^{o} - theta) = cos theta \$

\$cos(90^{o} - theta) = sin theta \$

\$tan(90^{o} - theta) = cot theta \$

\$cosec(90^{o} - theta) = sec theta \$

\$sec(90^{o} - theta) = cosec theta \$

\$cot(90^{o} - theta) = tan theta \$

\$sin (180° - θ) = sin θ\$

\$cos (180° - θ) = -cos θ\$

\$tan (180° - θ) = -tan θ\$

\$cosec (180° - θ) = cosec θ\$

\$sec (180° - θ) = -sec θ\$

\$cot (180° - θ) = -cot θ\$

\$sin (π/2 – θ) = cos θ\$

\$cos (π/2 – θ) = sin θ\$

\$sin (2π + θ) = sin θ\$

\$cos (2π + θ) = cos θ\$

\$sin (π/2 + θ) = cos θ\$

\$cos (π/2 + θ) = – sin θ\$

\$sin (π – θ) = sin θ\$

\$cos (π – θ) = – cos θ\$

\$sin (π + θ) = – sin θ\$

\$cos (π + θ) = – cos θ\$

\$sin (3π/2 – θ) = – cos θ\$

\$cos (3π/2 – θ) = – sin θ\$

\$sin (3π/2 + θ) = – cos θ\$

\$cos (3π/2 + θ) = sin θ\$

\$sin (2π – θ) = – sin θ\$

\$cos (2π – θ) = cos θ\$

\$sin(A+B) = sin A cos B + cos A sin B\$

\$sin(A-B) = sin A cos B - cos A sin B\$

\$cos(A+B) = cos A cos B - sin A sin B\$

\$cos(A-B) = cos A cos B + sin A sin B\$

\$tan(A+B) = {(tan A + tan B)} / {(1 - tan A tan B)}\$

\$tan(A-B) = {(tan A - tan B)} / {(1 + tan A tan B)}\$

\$cot(A + B) = {cot A cot B -1} / {cot B - cot A}\$

\$cot(A - B) = {cot A cot B + 1} / {cot B - cot A}\$

\$2 sin A⋅cos B = sin(A + B) + sin(A - B)\$

\$2 cos A⋅cos B = cos(A + B) + cos(A - B)\$

\$2 sin A⋅sin B = cos(A - B) - cos(A + B)\$

\$sinx⋅cosy = {sin(x + y) + sin(x − y)}/2\$

\$cosx⋅cosy = {cos(x + y) + cos(x − y)}/2\$

\$sinx⋅siny = {cos(x − y) − cos(x + y)}/2\$

\$sinx + siny = 2(sin((x + y)/2)cos((x − y)/2))\$

\$sinx − siny = 2(cos((x + y)/2)sin((x − y)/2))\$

\$cosx + cosy = 2(cos((x + y)/2)cos((x − y)/2))\$

\$cosx − cosy = −2(sin((x + y)/2)sin((x − y)/2))\$

\$sin^-1 (-x) = -sin^-1 x\$

\$cos^-1 (-x) = π - cos^-1 x\$

\$tan^-1 (-x) = -tan^-1 x\$

\$cosec^-1 (-x) = -cosec^-1 x\$

\$sec^-1 (-x) = π - sec^-1 x\$

\$cot^-1 (-x) = π - cot^-1 x\$

\$sin 2θ = 2 sinθ cosθ\$

\$cos 2θ = cos^2θ - sin^2θ\$

\$cos 2θ = 2 cos^2θ - 1\$

\$cos 2θ = 1 - 2 sin^2θ\$

\$cos 2θ = {1 - tan^2 θ}/{1 + tan^2 θ}\$

\$tan 2θ = {2 tanθ} / {1 - tan^2θ}\$

\$sec 2θ = sec^2 θ / {2 - sec^2 θ}\$

\$cosec 2θ = {sec θ. cosec θ} / 2\$

\$cot 2θ = {cot θ - tan θ}/2\$

\$sin(theta / 2) = +- sqrt { {1 - cos theta } / 2}\$

\$cos(theta / 2) = +- sqrt { {1 + cos theta } / 2}\$

\$tan(theta / 2) = +- sqrt { {1 - cos theta} / { 1 + cos theta } }\$

\$sin 3θ = 3sin θ - 4sin^3θ\$

\$cos 3θ = 4cos^3θ - 3cos θ\$

\$tan 3θ = (3tanθ - tan^3θ)/(1 - 3tan^2θ)\$