Converting Between Cartesian and Polar Coordinates

The video tutorial explains how to convert Cartesian functions to polar coordinates and vice versa. It begins with basic substitution using SOHCAHTOA principles and progresses through several examples, including converting y = 2x - 3 and more complex polar functions. The tutorial emphasizes algebraic manipulation and trigonometric identities to achieve these conversions, providing a toolkit for understanding the relationships between r, theta, x, and y. The video concludes with an abstract example, reinforcing the concepts and preparing viewers for graphing these functions.

What is the first step in converting the Cartesian equation 3y - 7x = 10 to polar coordinates?

• Substitute y with r cos(theta) and x with r sin(theta)

• Substitute y with r sin(theta) and x with r cos(theta)

• Factor out x and y

• Divide both sides by 10

After substituting y = r sin(theta) and x = r cos(theta) in the equation 3y - 7x = 10, what is the next step?

• Divide by sin(theta)

• Add 10 to both sides

• Multiply both sides by r

• Factor out r

In converting y = 2x - 3 to polar coordinates, what expression do you get after factoring out r?

• r(2 sin(theta) - cos(theta))

• r(sin(theta) - 2 cos(theta))

• r(sin(theta) + 2 cos(theta))

• r(2 sin(theta) + cos(theta))

What is the polar function given for conversion to Cartesian coordinates in the third section?

• r = 4 sin(theta)

• r = 4 cos(theta)

• r = sin(theta) + cos(theta)

• r = 2 sin(theta) + 3 cos(theta)

When converting r = 4 sin(theta) to Cartesian coordinates, what is the resulting equation?

• x^2 + y^2 = 4x

• x^2 + y^2 = 4

• x^2 + y^2 = 4y

• x^2 + y^2 = 16

In the conversion of r = sin(theta) + cos(theta) to Cartesian coordinates, what is the final form of the equation?

• x^2 + y^2 = x + y

• x^2 + y^2 = 2x + 2y

• x^2 + y^2 = y + x

• x^2 + y^2 = 2x - y

What is the Cartesian equivalent of the polar equation r = a^2?

• x^2 + y^2 = 4a^2

• x^2 + y^2 = 2a^2

• x^2 + y^2 = a^2

• x^2 + y^2 = a^4

What shape does the polar equation r = a^2 represent in Cartesian coordinates?

• Ellipse

• Parabola

• Circle

• Hyperbola

What is the significance of converting between polar and Cartesian coordinates?

• It simplifies complex numbers

• It allows for easier graphing of certain functions

• It is a method to solve quadratic equations

• It is used only in physics

Which trigonometric identities are primarily used in converting between polar and Cartesian coordinates?

• Sum and difference identities

• SOHCAHTOA

• Double angle identities

• Pythagorean identities