Understanding Spherical to Rectangular Coordinates

The video tutorial explains how to convert a spherical equation, where phi equals pi divided by six radians, into a rectangular equation. It begins by discussing the relationship between spherical and rectangular coordinates, specifically focusing on the angle phi. The tutorial then demonstrates the conversion process by substituting phi with its equivalent in rectangular terms, simplifying the equation through trigonometric identities, and finally deriving the rectangular equation of a cone. The video concludes by emphasizing the importance of restricting the z-value to non-negative numbers to accurately represent the cone's graph.

What is the angle phi in the given spherical equation?

• Pi divided by two radians

• Pi divided by three radians

• Pi divided by six radians

• Pi divided by four radians

What shape does the graph of the spherical equation form when rotated about the positive z-axis?

• A cone

• A cylinder

• A paraboloid

• A sphere

Which trigonometric function is used to relate phi to rectangular coordinates?

• Sine

• Tangent

• Cosine

• Secant

What is the result of taking the cosine of both sides of the equation?

• Sine of phi

• Tangent of phi

• Cosine of phi

• Arc cosine of phi

What is the value of cosine pi over six radians?

• 3/4

• Square root of 2 divided by 2

• Square root of 3 divided by 2

• 1/2

What operation is performed to eliminate the square roots in the equation?

• Taking the square root

• Squaring both sides

• Dividing by a constant

• Multiplying by a constant

What is the least common denominator used to clear fractions in the equation?

• 4

• 2

• 5

• 3

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

• x squared plus y squared equals z squared

• 3x squared plus 3y squared equals z squared

• x squared plus y squared plus z squared equals 1

• 3x squared plus 3y squared plus 3z squared equals 4z squared

What restriction is applied to the rectangular equation to ensure it represents the correct graph?

• x is greater than or equal to zero

• y is greater than or equal to zero

• z is greater than or equal to zero

• z is less than or equal to zero

What does the final graph of the equation represent?

• A paraboloid

• A cylinder

• A cone

• A sphere