Question

Which of the following best explains why a complex number z = a + bi and its polar form represent the same number?

1) By definition, cosine (θ) = r/a and sine (θ) = r/b.
2) By definition, cosine (θ) = r/b and sine (θ) = r/a.
3) By definition, cosine (θ) = a/r and sine (θ) = b/r.
4) By definition, cosine (θ) = b/r and sine (θ) = a/r.

Answer 1

A complex number z = a + bi and its polar form represent the same point; they are related by the equations cosine (θ) = a/r and sine (θ) = b/r, which provide a connection between Cartesian and polar coordinates.

Step-by-step explanation:

The question asks why a complex number z = a + bi and its polar form represent the same number. The correct statement that explains this is option 3) By definition, cosine (θ) = a/r and sine (θ) = b/r. This is because in polar coordinates the point P (represented by the complex number z) has a magnitude r (the distance from P to the origin) and an angle θ with the x-axis. The components of P in the Cartesian coordinate system are a and b, corresponding to the x and y coordinates, respectively. Hence, in the relationship between Cartesian and polar coordinates, we use trigonometric functions to project r onto the x- and y-axes resulting in a = r cos θ and b = r sin θ.