Law of Sines

SinA / a = SinB / b = SinC / c

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