Home » Polar Coordinates

# Convert complex number in polar form to the form a + bi

Convert each of the following complex numbers given in polar form to the form .

1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .

1. Using the definition of the complex exponential () we have

2. Again, using the definition of the complex exponential we have,

3. We have

4. We have

5. We have

6. We have

7. We have

8. We have

# Sketch a “limacon” and compute its area from 0 to 2 π

Define a limacon by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch a “cardioid” and compute its area from 0 to 2 π

Define a cardioid by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

where we know from this exercise (Section 2.11, Exercise #7).

# Sketch a “four-leaf clover” and compute its area from 0 to 2 π

Define a four-leaf clover by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

where for the final step we used the previous exercise (Section 2.11, Exercise #12).

# Sketch a “lazy eight” and compute its area from 0 to 2π

Define a lazy eight by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch a “four-leaved rose” and compute its area from 0 to 2π

Define a four-leaved rose by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch a “rose petal” and compute its area from 0 to π/2

Define a rose petal by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch two circles tangent to the x-axis and compute their area from 0 to 2π

Define two circles tangent to the -axis by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch a circle tangent to the x-axis and compute its area from 0 to π

Define a circle tangent to the -axis by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch two circles tangent to the y-axis and compute their area from 0 to 2 π

Define two circles tangent to the -axis by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,