coordinate geometry and trig identities

what is the 3d and 2d equation for a circle

(x-a)^2+(y-b)^2+(z-c)^2=r^2

(x-a)^2+(y-b)^2=r^2

converting degrees to radians

\frac{180}{\pi}*radians = degrees

\frac{\pi}{180}*degrees=radians

what are polar coordinates

→ how is it written

describes position using an angle \theta anticlockwise and a distance r from a center point

(r,\theta)

what are cartesian coordinates

describes position using XYZ values

how do you convert from cartesian ←→ polar coordinate

cartesian → polar

r=\sqrt{x^2+y^2}

\theta=tan^-1 (\frac{y}{x})

polar → cartesian

x= r~cos(\theta)

y=r~sin(\theta)

what are cylindrical coordinates

→ how is it written

describes a 3D position using:

1. anticlockwise angle from an origin: \theta~or~\varphi

2. radius r or \rho

3. height: z

(r,\theta,z) or (\rho,\varphi,z)

cartesian ←→ cylindrical

cartesian → cylindrical:

r=\sqrt{x^2+y^2}

\theta=tan^-1 (\frac{y}{x})

Z = Z

cylindrical → cartesian:

x= r~cos(\theta)

y=r~sin(\theta)

Z = Z

what are spherical coordinates?

describes a 3D position depending on:

1. radius: r

2. zenith/elevation angle: \theta

3. azimuthal angle (flat ange): \varphi

(r,\theta,\varphi)

cartesian ←→ spherical

cartesian → polar

r=\sqrt{x^2+y^2+z^2}

\theta=cos^{-1}(\frac{z}{r})

\varphi=tan^{-1}(\frac{y}{x})

spherical → cartesian

z=r~cos(\theta)

x=r~sin(\theta)cos(\varphi)

y=r~sin(\theta)sin(\varphi)

what is a solid angle?

→ how is it calculated

→ what are the units?

solid~angle (\Omega) =\frac{surface~area~of~object}{radius~of~unit~sphere^2} REMEMBER THE UNIT CIRCULE RADIUS IS SQUARED

→ measured in steradians

what are the assumptions when calculating unit circle

1. the object you are observing can be considered as a circle of which you can calculate the surface area of (

2. you are standing at the center of the unit circle (apex)

what it tan

\frac{sin}{cos}

what are sec, csc, and cot

sec=\frac{1}{cos} csc=\frac{1}{sin} cot=\frac{1}{tan} = \frac{cos}{sin}

what are the power of trig identities

cos^2(x)+sin^2(x)=1

sec^2=1+tan^2(x)

csc^2=cot^2+1

these are based on pythagorea’s theoremw

what are the double angle trig identities

sin(2X)=2(cos(X)sin(X))

cos(2x) = cos^2(x)-sin^2(X) = 2cos^2(x)-1 = 1-2sin^2(x)

tan(2x)=\frac{2tan(x)}{1-tan²(X)}

half-angle trig formulas

sin(\frac{x}{2})=\pm\sqrt{\frac{1-cos(x)}{2}}

\cos(\frac{x}{2})=\pm\sqrt{\frac{1+cos(x)}{2}}

\tan(\frac{x}{2})=\pm\frac{1-cos(x)}{\sin\left(x\right)}

sum and differences

sin(a\pm b)=sin(a)cos(b)\pm cos(A)sin(b)

\cos(a\pm b)=\cos(a)cos(b)\pm\sin(A)sin(b)

tan(a\pm b)=\frac{tan(A)\pm tan(b)}{1\pm tan(A)tan(B)}

product to sum

sin(A)sin(b)=\frac{1}{2} (cos(a-b)-cos(a+b))

\cos(A)\cos(b)=\frac{1}{2}(cos(a-b)+cos(a+b))

\sin\left(A)\cos(b\right)=\frac{1}{2}(\sin(a+b)+\sin(a-b))

\cos(A)sin(b)=\frac{1}{2}(\sin(a+b)-\sin(a-b))

sum to product