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SinA / a = SinB / b = SinC / c
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Law of Cosines
a² = b² + c² - 2bc * cosA
Set of all points where z = 0 is called….
The xy - plane
Set of all points where x = 0 is called…
The zy - plane
Set of all points where y = 0 is called…
The xz - plane
Coordinate Planes
The xy, yz, and xz planes
Distance Formula
dis(P, Q) = √(x1 - x2)² + (y1 - y2) + (z1 - z2)
Midpoint Formula
Take the average of each coordinate:
p + q / 2
Polar Coordinates
(r,θ)
θ (for polar)
Distance off of the positive x-axis
r (radius) in polar
Distance from the origin
Cartesian/Rectangular Coordinates
(x,y,z)
Converting Polar and Cartesian
x = rcosθ
y = rsinθ
r² = x² = y²
θ = tan-1(y/x)
Cylindrical Coordinates
(r, θ, z)
r in Cylindrical Coordinate
Distance to the z- axis
Spherical Coordinates
(P, θ, Φ)
(Rho, Theta, Phi)
P (Rho)
(sph) Distance to the origin
θ
(Same as cylindrical) angle off of the positive x-axis in xy-plane
Φ (Phi)
tilt angle off of the positive z-axis
Sphere Equation (centered at origin) in cartesian
x² + y² + z² = p²
Sphere Equation (centered at origin) in cylindrical
r² + z² = p²
Sphere Equation (centered at origin) in spherical
p = a
Circle equation (centered at origin) in cartesian
x² + y² = r²
Circle equation (centered at origin) in polar
r = a
The length of one wavelet is…
The period
Amplitude
The maximum height the curve reaches
Formula for Period
2π/b
Standard Form of Sinusoidal Curves
y = asin(bx)
y = acos(bx)
*(a = amplitude)
y = af(x)
a > 1 means…
Vertical Stretch
y = af(x)
0 < a < 1 means…
Vertical Shrink
y = f(bx)
b > 1 means…
(big #)
Horiztonal Shrink
y = f(bx)
0 < b < 1 means….
(small #)
Horizontal Stretch
Parabola
The set of all points that are equidistant from the focus point and the directrix line
Standard Form of a Parabola
±4c(y-k) = (x-h)²
c means…
(parabola)
distance from the vertex to the focus/directrix
The vertex comes from…
(h,k)
h tells us…
(parabola)
The x-value of focus and vertex and the AOS
± tells us ….
(parabola equation)
(-) = parabola opens down
(+) = parabola opens up
Ellipse
The set of all points where the sum of distances between foci is constant
Standard Form of an Ellipse:
(x-h)²/a² + (y-k)²/b² = 1