Polar Coordinates

Front

What are polar coordinates?

Back

Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis).

Front

How do you convert rectangular coordinates (x, y) to polar coordinates (r, θ)?

Back

To convert (x, y) to (r, θ): 1. Calculate r = √(x² + y²). 2. Calculate θ = arctan(y/x).

Front

What is the absolute value of a complex number a + bi?

Back

The absolute value of a complex number a + bi is given by |a + bi| = √(a² + b²).

Front

In which quadrant does the point (x, y) lie if x > 0 and y > 0?

Back

The point (x, y) lies in Quadrant 1.

Front

What is the polar coordinate representation of the point (0, 0)?

Back

The polar coordinate representation of the point (0, 0) is (0, θ) for any angle θ.

Front

How do you find the angle θ in polar coordinates when given the rectangular coordinates?

Back

The angle θ can be found using θ = arctan(y/x), adjusting for the correct quadrant.

Front

What is the polar coordinate of the point (6√3, 6)?

Back

The polar coordinate of the point (6√3, 6) is (12, π/6).

Front

What is the product of two complex numbers (a + bi)(c + di)?

Back

The product is given by (ac - bd) + (ad + bc)i.

Front

What is the significance of the angle in polar coordinates?

Back

The angle in polar coordinates indicates the direction of the point from the origin.

Front

How do you determine the quadrant of a point given in polar coordinates (r, θ)?

Back

The quadrant can be determined by the signs of r and the angle θ: 1. Quadrant 1: r > 0, 0 < θ < π/2; 2. Quadrant 2: r > 0, π/2 < θ < π; 3. Quadrant 3: r > 0, π < θ < 3π/2; 4. Quadrant 4: r > 0, 3π/2 < θ < 2π.