Understanding Rectangular and Spherical Coordinates

The video tutorial explains how to convert a spherical equation, rho equals two cosecant phi, into a rectangular equation using x, y, and z coordinates. It involves substituting trigonometric identities, squaring the equation, and simplifying to eliminate spherical variables. The final rectangular equation, x squared plus y squared equals four, represents a cylinder in three-dimensional space.

What is the primary goal when converting a spherical equation to rectangular coordinates?

• To include spherical variables

• To eliminate spherical variables

• To change the equation's shape

• To simplify the equation

Which trigonometric identity is used to replace cosecant in the equation?

• Cosecant equals one over cosine

• Cosecant equals cosine over sine

• Cosecant equals one over sine

• Cosecant equals sine over cosine

Why is it easier to substitute for cosine phi rather than sine phi in this context?

• Because cosine phi is simpler

• Because z equals rho times cosine phi

• Because sine phi is not used

• Because cosine phi is more common

What is the result of squaring both sides of the equation?

• Rho squared equals four times sine squared phi

• Rho squared equals four divided by sine squared phi

• Rho squared equals four divided by cosine squared phi

• Rho squared equals four times cosine squared phi

Which identity is used to replace sine squared phi?

• Sine squared phi equals one minus sine squared phi

• Sine squared phi equals one plus cosine squared phi

• Sine squared phi equals one minus cosine squared phi

• Sine squared phi equals cosine squared phi

How is cosine phi expressed in terms of z and rho?

• Cosine phi equals rho divided by z

• Cosine phi equals z divided by rho

• Cosine phi equals z times rho

• Cosine phi equals rho times z

What is the next step after substituting z squared divided by rho squared for cosine squared phi?

• Divide both sides by the numerator

• Subtract the fractions

• Multiply both sides by the denominator

• Add the fractions

What substitution is made for rho squared in the final steps?

• Rho squared equals y squared minus z squared

• Rho squared equals x squared plus y squared

• Rho squared equals x squared plus y squared plus z squared

• Rho squared equals x squared minus y squared

What is the final rectangular equation derived from the spherical equation?

• x squared plus y squared equals four

• y squared plus z squared equals four

• x squared minus y squared equals four

• x squared plus z squared equals four

What geometric shape does the final rectangular equation represent?

• A cylinder

• A plane

• A sphere

• A cone