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H Pre-Calculus Semester 1 Final Review
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Gravity
Key Concepts:
Terms in this set (45)
S = rθ
Arc Length Formula in radians
S = πrθ /180
Arc Length Formula in degrees
sinA/a=sinB/b=sinC/c
Law of Sines
less than the height
No triangle when s2 is...
equal to the height
One triangle when s2 is...
greater then the height
Two triangles when s2 is...
a²=b²+c²-2bcCosA
Law of Cosines
Angles that share a terminal and initial side
Coterminal Angle
Acute angle between the terminal side and the x-axis
Refrence Angle
2π
Period for sin, cos, sec, and csc
π
Period for tan and cot
Half of the height |a|
Amplitude Equation and Definition
T = 2π/|b|
Period Equation
sinθ = 1/cscθ ; cscθ = 1/sinθ
cosθ = 1/secθ ; secθ = 1/cosθ
tanθ = 1/cotθ ; cotθ = 1/tanθ
Reciprocal Identities
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ
Quotient Identities
sin^2x+cos^2x=1
1+tan^2x=sec^2x
1+cot^2x=csc^2x
Pythagorean Identities
sin (π/2 - x) = cos x
cos (π/2 - x) = sin x
tan (π/2 - x) = cot x
cot (π/2 - x) = tan x
sec (π/2 - x) = csc x
csc (π/2 - x) = sec x
Cofunction Identities
sin(-x) = - sin x
cos(-x) = cos x
tan (-x) = - tan x
csc (-x) = - csc x
sec (-x) = sec x
cot (-x) = - cot x
Odd-Even Identities
sin 2u = 2 sin u cos u
cos 2u = cos^2 u- sin^2u
Double Angle Identities
sin^2u=(1-cos2u)/2
cos^2u=(1+cos2u)/2
tan^2u=(1-cos2u)/(1+cos2u)
Power Reducing Identities
cos (u +/- v) = cos u cos v +/- sin u sin v
Cosine Sum and Difference Identity
sin (u +/- v) = sin u cos v +/- sin v cos u
Sine Sum and Difference Identity
tan (u + v) = (sin u cos v + sin v cos u) / (cos u cos v - sin u sin v)
tan (u - v) = (sin cos v - sin v cos u) / (cos u cos v + sin u sin v)
Tangent Sum and Difference Identity
|v| = Square root of (x2-x1)^2 + (y2-y1)^2
Magnitude Formula
Unit Vector
Vector with a magnitude of 1.
u = v/|v|
Direction Angles a equation
a = |v| cosθ
Direction Angles b equation
b = |v| sinθ
i = < 1 , 0 >
j = < 0 , 1 >
Unit vectors i and j
u dot v= u1
v1 + u2
v2
Dot Product
θ = cos^-1 ( u dot v/ |u| |v| )
Angle Between Vectors Formula
It means they are perpendicular and u dot v = 0
What does it mean when two vectors are orthoganol?
projvU = (u dot v/|v|^2)v
Projection of u onto v
W = |F| |AB|
Work if F has the same direction as AB
W = |F| |AB| cosθ
Work if θ is the angle between F and AB
y = -16t^2 + ( v sinθ) t + v
Equation for Gravity
x = rcosθ
y = rsin θ
r^2 = x^2 + y ^2
tanθ = y/x
Polar Coordinate Conversions
Replace ( r, θ ) with (r, -θ) or ( -r, π-θ)
Symmetry about the x-axis
Replace ( r, θ ) with (-r, -θ) or ( r, π-θ)
Symmetry about the y-axis
Replace ( r, θ ) with (-r, θ) or (r, π + θ)
Symmetry about the origin
r = a cos n θ
r = a sin n θ
n > 1
Equation for a Rose Curve
n even: x-axis, y-axis, and origin
n odd and cosine: x-axis
n odd and sine: y-axis
Symmetry for Rose Curves
r = a +/- b sin θ
r = a +/- b cos θ
Limaçon Curve Equation
innerloop: a/b < 1
cardiod: a/b = 1
dimpled: 1 < a/b < 2
convex:; a/b < 2
Types of Limaçons and how to calculate it
r = θ
Spiral of Archimedes
r^2 = a^2 cos 2θ
r^2 = a^2 sin 2θ
Lemniscate Curve
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