Pre-Calc Formulas

Inclination of a line

m = tan𝜃

Angle between two lines

tan𝜃 = |m₂-m₁| / |1+m₁m₂|

Distance between a point and a line

d = |Ax₁+By₁+C| / (sqrt/A²+B²)

Sum of an infinite geometric series

S = a₁/1-r

Sum of a finite geometric series

S_n = a₁((1-rⁿ)/(1-r))

Form of the nth term of a geometric sequence

a_n = a₁r^n-1

Sum of a finite arithmetic sequence

S_n = n/2(a₁+a_n)

Form of the nth term of an arithmetic sequence

a_n=a₁+(n-1)d

Polar to Rectangular coordinate conversion formulas

x = rcos𝜃 y=rsin𝜃

Rectangular to polar coordinate conversion formulas

tan𝜃 = y/x r²=x²+y²

Equivalent polar coordinates

(r, 𝜃 ) = (2, 𝜃 ± 2nπ). ( r, 𝜃) = (-r, 𝜃 ± (2n+1)π)

Formulas for rotating conics

cot(2𝜃) = (A-C)/B x = Xcos𝜃 - Ycos𝜃 y = Xcos𝜃 + Ycos𝜃

Invarients under conic rotatiom

F = F’. A+C = A’+C’. B² - 4AC = (B’)² - 4AC

Classification of conics by discriminant

B² - 4AC < 0, ellipse or circle. B² - 4AC = 0, parabola. B² - 4AC > 0, hyperbola.

Classifying conics by general equation

Circle: A=C. Parabola: AC = 0 (but not both). Ellipse: AC> 0. Hyperbola AC<0.

Equations for hyperbolas

standard: (trans axis is horizontal, switch a and b for vertical)

(x-h)²/a² - (y-k)²/b²= 1. a = center to vertex, c = center to foci. c² = a² + b²

asymptotes (horizontal, switch a and b for vertical) y =k±b/a(x-h)

Equations for ellipses

Standard equation: (when major axis is horizontal, switch a and b for vertical) (x-h)²/a²+(y-k)²/b²=1. major axis = 2a, minor axis = 2b. c is distance from foci to center. c²= a² - b²

Eccentricity

e = c/a

Equations for parabolas

Standard equation: (vertical axis, switch x and y for horizontal)

(x-h)²=4p(y-k). Focus is on axis, p units from vertex. Directrix: y=k-p (if horizontal, sub y for x and k for h)

Tangent line properties

1. Makes equal angles with line from axis-focus and the axis.

2. d from P to focus = distance from focus to where tangent intersects axis

Inclination of a line formula

m = tanθ

Angle between two lines formula

tanθ=|m₂-m₁/1+m₁m₂|

Distance between a point and a line formula

d = |Ax₁ + By₁+ C| / sqrt/A²+B²

Cramer’s rule formula

If a₁b₂-a₂b₁ doesnt =0, x = c₁b₂-c₂b₁/a₁b₂-a₂b₁. y = a₁c₂-a₂c₁/a₁b₂-a₂b₁

Inverse of a 2×2 matrix

If A is a 2×2 matrix given by: A = [ a b c d ] then A^-1 = 1/ad-bc[d -b -c a]

Simple interest formula

I = Prt

Abs value of a complex number

| ai + b | = sqrt/a²+b²

Trig form of a complex # (lots of formulas)

Z = r(cosθ + isinθ) where r is the abs value of the complex number

Multiplication and division of complex numbers in trig form

Z₁Z₂ = r₁r₂[cos(θ₁+θ₂) + isin(θ₁+θ₂)] Z₁Z₂ = r₁/r₂[cos(θ₁- θ₂) + isinθ₁- θ₂)]

Power of a complex number

Zⁿ = [r(cosθ+isinθ)]ⁿ = rⁿ(cos(nθ) + isin(nθ))

Roots of complex numbers

u = a + bi is an nth root of Z is Z = uⁿ = (a+bi)ⁿ. There are n distinct nth roots given by nthrt/r (cos((θ+2piK)/n) + isin((θ+2piK)/n)) Where K = 0, 1, … , n-1

The dot product of u = {u₁, u₂} and v = {v₁, v₂}

u . v = u₁v₁ + u₂v₂

Angle between two vectors and orthogonal vectors

cosθ = u.v / ||u|| x ||v|| for orthogonal vectors, u.v=0

Vector components and projection

W₁ and W₂ are vector components. u and v are nonzero vectors. u = W₁ + W₂, components are orthogonal, W₁ is parallel to v, W₁ is the projection of u onto v and is denoted by: W₁ = proj_vu. Projection of u onto v is

Proj_vu = (u.v/||v||²)V. To find force, project F onto V in direction of ramp

Work formula, vectors

W = (magnitude of force)(distance) = ||F||||→ PQ||

Polar equations of conics and what they mean

r = ep/1 ± ecosθ when directrix is vertical (if -, to the left, if +, to the right). r = ep/1 ± esinθ when the directrix is vertical (if - below pole, if + above pole). e = eccentricity, directrix: y/x = p. vertex is in middle of directrix and pole.

polar forms of different graphs

for r = a ± bsinθ or r = a ± bcosθ: if a/b<1 limacon w/ loop,if a/b=1 cardioid, 1<a/b<2 dimpled limacon, a/b> 2, convex limacon. for r=acos(nθ) or r=asin(nθ) rose curve with n petals if n is odd or 2n petals if n is even. Petals are length a. r=acosθ or sin is a circle. r²=a²sin2θ is a diagonal lemniscate with length a, r²=a²cos2θ is a horizontal lemniscate with lenth a

DeMoivre’s Theorem

if Z is a complex number and n is an integer, $Z^{n}=\left(r\left(\cos\theta+i\sin\theta\right)\right)^{n}$ or $r^{n}\left(\cos n\theta+i\sin n\theta\right)$

Roots of complex numbers$Z=r\left(\cos\theta+i\sin\theta\right)$

Has n distinct roots given by $n\sqrt{r}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)$ Where k=0,1,2…,n-1