Converting Between Spherical and Rectangular Equations

This video tutorial covers the conversion between spherical and rectangular coordinate systems. It begins with an introduction to spherical coordinates, explaining the roles of rho, theta, and phi. The tutorial then demonstrates how to convert a rectangular equation, x=2, into spherical form, followed by converting spherical equations back to rectangular form. The video includes examples and applications, such as converting equations of planes and spheres, and discusses the use of trigonometric identities and unit circle properties in these conversions.

In spherical coordinates, what does the variable rho represent?

• The distance from the x-y plane

• The angle from the positive z-axis

• The distance from the origin

• The angle from the positive x-axis

What is the spherical form of the equation x = 2?

• rho = 2 cos(phi) sin(theta)

• rho = 2 / (sin(phi) cos(theta))

• rho = 2 tan(phi) sec(theta)

• rho = 2 sin(phi) cos(theta)

Which trigonometric identity is used to simplify 1/sin(phi)?

• Cosecant

• Secant

• Cotangent

• Tangent

What is the spherical form of the equation x^2 + y^2 - z^2 = 0?

• rho^2 = x^2 + y^2 + z^2

• cos^2(phi) = 1/2

• rho^2 = 2z

• rho = 2 cos(phi)

What is the cosine value for phi when solving the equation for phi?

• 1

• -sqrt(2)/2

• 0

• sqrt(2)/2

What is the rectangular form of the equation theta = pi/4?

• x = y

• x = 2y

• y = 2x

• y = x

In the conversion of rho = 2 cos(phi) to rectangular form, what is the resulting equation?

• x^2 + y^2 = z^2

• x^2 + y^2 + z^2 = 2z

• x^2 + y^2 = 4z

• x^2 + y^2 + z^2 = 4z

What is the center of the sphere in the equation x^2 + y^2 + (z-1)^2 = 1?

• (0, 0, 1)

• (1, 0, 0)

• (1, 1, 1)

• (0, 0, 0)

What is the radius of the sphere in the equation x^2 + y^2 + (z-1)^2 = 1?

• 2

• 1/2

• 1

• sqrt(2)

What is the standard form of a sphere's equation derived from rho = 2 cos(phi)?

• x^2 + y^2 + z^2 = 1

• x^2 + y^2 + z^2 - 2z = 0

• x^2 + y^2 + z^2 + 2z = 0

• x^2 + y^2 + z^2 = 4