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pythagorean identity
sin²θ+cos²θ=1
cofunction identities
sinθ=cos(pi/2-θ), etc.
meaning, a trig function with no co in front (sine, tangent, secant) of theta is equal to its co-equivalent of 45deg-θ. this works the other way around too :)
cos(A+B)
cosAcosB - sinAsinB
sin(A+B)
sinAcosB+sinBcosA
tan(A-B)
(tanA)+(tanB)/(1-(tanA*tanB))
cos(A-B)
cosAcosB-sinAsinB
sin(A-B)
sinAcosB-sinBcosA
tan(A-B)
(tanA)+(tanB)/(1+(tanAtanB))
sin(2θ)
2sinθ*cosθ
cos2θ in terms of sin?
1-2sin²θ
cos2θ in terms of cosine
2cos²θ-1
cos2θ in terms of cosine and sine
cos²θ-sin²θ
tan2θ
2tanθ/(1-tan²θ)
sin(θ/2)
±√(1-cosθ)/2)
cos(θ/2)
±√((1+cosθ)/2)
tan(θ/2) in terms of cosine
±sqrt(1-cosθ/1+cosθ)
tan(θ/2) in terms of sine and cosine
sinθ/1+cosθ
tanθ/2 in terms of cosine and sine
1-cosθ/sinθ
sin²θ
1-cos2θ/2
cos²θ
1+cos2θ/2
tan²θ
1-cos2θ/1+cos2θ
sinAsinB
1/2(cos(A-B)-cos(A+B))
cosAcosB
1/2(cos(A+B)+cos(A-B))
sinAcosB
1/2(sin(A+B) + sin(A-B))
cosAsinB
1/2(sin(A+B)-sin(A-B))
sinA+sinB
2*sin(A+B/2)*cos(A-B)/2)
cosA+cosB
2*cos(A+B/2)*cos(A-B/2)
cosA-cosB
-2*sin(A+B/2)*sin(A-B/2)
sinA-sinB
2*cos(A+B/2)*sin(A-B)/2
harmonic motion
acoswt/asinwt
law of sines
sinA/a = sinB/b = sinC/c
law of cosines
cosA=b²+c²-a²/2bc.
the cosine of a given angle is the square of its non-opposite sides minus its opposite side squared over 2 times the opposite side.
heron’s formua
given s = ((a+b+c)/2, A=sqrt(s(s-a)(s-b)(s-c))
conversion from polar to cartesian coordinates
x=rcosθ
y=rsinθ
r² = x²+y²
cosθ=x/r, sinθ=y/r
conversion from cartesian to polar
r = sqrt(x²+y²)
θ=arctan(y/x)
standard polar form of a circle
r=asinθ or r=acosθ
polar form of a cardioid
a±bcosθ/a±bsinθ given that a/b=1
one loop limacon
a±bcosθ/a±sinθ when a>0, b>), and a/b is between one and two
inner loop limacon
a±bcosθ/a±bsinθ when a>0,b>0, and a<b
lemniscate (weird infinity)
r²=a²cos2θ/r²=a²sin2θ, a=/= 0
when a is negative with cosine it looks like a peanut
when a positive with sine its rotated peanut
rose curve
r=a(cos or sin)nθ.
if n is even, there are 2n “petals”
if n is odd, there are n petals
archimedes spiral
r=θ
θ>0
also occurs when n/θ or nθ²+ncosθ
parametric equation
a way of defining an equation for an x and y coordinate using t. graphed in the form x=(whatever*t),y=(whatever*t). has arrows to tell you what the graph is doing as t increases
parameterizing a curve
a way of converting a curve (or basically any equation) in the form of y=x to being in terms of t
determine an equation for x in terms of t and substitute that into the y= equation. i like to use x=t, but some questions and situations require you to be quirky :P
parameterizing a linear path given two points
find the rate of change for x in terms of t, write an equation in the form x(t) = ROCt+(initial x-value). repeat for y
eliminating the parameter
solve one equation for t and plug in the result to the other
parametric equation of a projectile
x=(initialvelocity*cosθ)*t
y=-1/2gt²+(intialvelocity*sinθ)*t+h
force of gravity is either 9.8 m/s² or 32.2ft/s²
circular motion on a ferris wheel
x(t) = rcos(ωt) = 50cos(ωt)
y(t) = r·sin(ωt) = 50sin(ωt)
omega is angular speed (remember her…!)
magnitude of a vector
sqrt(a²+b²) in a vector <a,b>
direction of a vector
arctan(b/a)
sum of vectors <u1,u2> and <v1,v2>
<u1+v1, u2+v2>
unit vector of the same direction as vector v.
v/|v|
dot product of vectors <a,b> <c,d>
u*v = ac+bd
formula of the angle between two vectors
cos(θ) = u/|u| * v/|v|
standard form of a velocity vector
<|v|cosθ,|v|sin)θ>
how would one find the displacement of an object with several vectors acting upon it?
add the vectors all together
how would one find the “speed” of an object represented by a vector after being acted upon by another vector, such as the speed of a boat when acted upon by a current?
take the magnitude of the resulting vector after finding the displacement
what is an inconsistent system of equations?
a system that has no solution, usually meaning that the equations are paralle.
what is a dependent system?
a system that has infinitely many solutions, meaning the equations are the same.
what is partial fraction decomposition?
taking a fraction containing variables and splitting it up into its most basic parts being added together
What is the formula for partial fraction decomposition with nonrepeating linear factors?
A/(factor 1) + B/(factor two) … etc.
What is the form of partial fraction decomposition with repeating linear factors?
A/(ax+b) + B/(ax+b)² … etc
what is the form of fraction decomposition when there are nonrepeating quadratic factors?
Ax+B/(factor one) + Cx+D/(factor two) … etc
What is the form of fraction decomposition when the denominator contains repeating quadratic factors?
Ax+B/(ax+b+c) + Cx+D/(ax+b+c)² etc
what must be true about matrices for them to be multiplied?
the amount of columns in matrix A must be equal to the amount of rows in matrix B
how does one multiply a matrix?
write matrix A out. transpose matrix B above it, and multiply each term by its corresponding term (this only makes sense to me)
do matrices follow the commutative property?
no! AB does not equal BA, as one or the other may not exist. but they also just dont
what is gaussian elimination?
the process by which a system of equations with three variables can be solved using matrices. it requires getting the square matrix into row-echelon form.
what is row-echelon form?
a way of arranging a square matrix that has each entry have only zeroes below it.
how does one find the inverse of a 2×2 matrix?
multiply the reciprocal of the determinant by the matrix with its top left and bottom right entries swapped and its top right and bottom left entries’ signs changed
what is cramer’s rule for 2×2 systems?
x= D(x)/D, y = D(y)/D
what is cramer’s rule for 3×3 systems?
x=D(x)/D, y=D(y)/D, z=D(z)/D. Dx = dbc, Dy = adc, Dz = abd, given that the starting equation is wrriten as "ax+by+cz=d”
the variable’s column is replaced by the constant’s column
standard form for an ellipse with a major axis of x
(x-h)²/a²+(y-k)²/b²=1
vertices at (h±a, k). vertices always occur on the major axis
foci at (h±c,k)
ellipse with a major axis of y
(x-h)²/b²+(y-k)/a² =1
vertices at (h, k±a). vertices always occur on the major axis.
foci at (h,k±c)
hyperbola (opens horizontally)
(x-h)²/a² - (y-k)²/b²=1
vertices at (h±a,k)
foci at (h±c,k)
asymptotes at y=±b/a(x-h)+k
hyperbola (opens upwards)
(y-k)²/a²-(x-h)²/b²=1
vertices on major axis
foci at (k,h±c)
asymptote at y=±a/bx
equation of a vertical parabola
(x-h)² =4p(y-k)
focus: (h,k±p)
directrix: y=k-p
equation of a horizontal parabola
(y-k)²=4p(x-h)
focus at (h±p,k)
x=h-p
Ax²+Bxy+Cy²+Dx+Ey+F=0, AC =0. What is the conic?
Without completing the square, the conic is a parabola.
Ax²+Bxy+Cy²+Dx+Ey+F=0, AC >0. What is the conic?
Without completing the square, this conic is either a circle or an ellipse.
Ax²+Bxy+Cy²+Dx+Ey+F=0, AC < 0. What is the conic?
Without completing the square, the conic is a hyperbola.
Ax²+Bxy+Cy²+Dx+Ey+F=0, B²-4AC =0
Using the discriminant, the conic is a parabola
Ax²+Bxy+Cy²+Dx+Ey+F=0, B²-4AC < 0
Using the discriminant, the conic is a hyperbola
Ax²+Bxy+Cy²+Dx+Ey+F=0, B²-4AC > 0
Using the discriminant, the conic is either a circle or an ellipse.
What is eccentricity?
The distance from a given point to the focus divided by the distance from the same given point to a given line.
What does it mean if the eccentricity is between zero and one?
If the eccentricity is in this interval, the conic is a parabola.
What does it mean if the eccentricity is 1?
When the eccentricity is this value, the conic is a parabola.
What does it mean if the eccentricity is less than one?
When the eccentricity is below this value, it means the conic is a hyperbola.
Polar equations for a conic.
r=ep/(1±ecosθ) / r=ep/(1±esinθ). What
What is the equation of the directrix of a polar conic given its denominator includes cosine?
x=±p
What is the equation of the directrix of a polar conic given its denominator includes sine?
y=±p
Recursively defined linear sequence.
(an-1)+d
Explicitly defined linear sequence
a(1) + d(n-1)
recursively defined geometric sequence
d(an-1)
explicitly defined geometric sequene
(an)(d)n-1
Formula for the sum of the first n terms of an arithmetic sequence
n(a1+an)/2
Formula for the sum of the first n terms of a geometric sequence
a1-a1r^n/1-r
Given the common ratio is less than one, what is the approximate sum of the infinite geometric sequence?
a1/1-r
What is the value of 0!
Weird factorial equal to one.
What a permutation?
The number of possible arrangements, given that the order of items matters. (1,2,3 is a different arrangement than 3,1,2)
Note
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