### All Precalculus Resources

## Example Questions

### Example Question #1 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

in rectangular form.

**Possible Answers:**

**Correct answer:**

Using the general form of a polar equation:

we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations.

Distributing the 4, we obtain the final answer of:

### Example Question #5 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

in rectangular form.

**Possible Answers:**

**Correct answer:**

Using the general form of a polar equation:

we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations.

distributing the 5, we obtain the final answer of:

### Example Question #6 : Express Complex Numbers In Rectangular Form

Convert in rectangular form

**Possible Answers:**

**Correct answer:**

To convert, just evaluate the trig ratios and then distribute the radius.

### Example Question #7 : Express Complex Numbers In Rectangular Form

Convert to rectangular form

**Possible Answers:**

**Correct answer:**

To convert to rectangular form, just evaluate the trig functions and then distribute the radius:

### Example Question #8 : Express Complex Numbers In Rectangular Form

Convert to rectangular form

**Possible Answers:**

**Correct answer:**

To convert, evaluate the trig ratios and then distribute the radius:

### Example Question #1 : Express Complex Numbers In Polar Form

The following equation has complex roots:

Express these roots in polar form.

**Possible Answers:**

**Correct answer:**

Every complex number can be written in the form **a + bi**

The polar form of a complex number takes the form r(cos + isin )

Now r can be found by applying the Pythagorean Theorem on a and b, or:

**r = **

can be found using the formula:

=

So for this particular problem, the two roots of the quadratic equation

are:

Hence, a = 3/2 and b = 3√3 / 2

Therefore r = ** = **3

and = tan^-1 (√3) = 60

And therefore x = r(cos + isin ) = 3 (cos 60 + isin 60)

### Example Question #2 : Express Complex Numbers In Polar Form

Express the roots of the following equation in polar form.

**Possible Answers:**

**Correct answer:**

First, we must use the quadratic formula to calculate the roots in rectangular form.

Remembering that the complex roots of the equation take on the form a+bi,

we can extract the a and b values.

We can now calculate r and theta.

Using these two relations, we get

. However, we need to adjust this theta to reflect the real location of the vector, which is in the 2nd quadrant (a is negative, b is positive); a represents the x-axis in the real-imaginary plane, b represents the y-axis.

The angle theta now becomes 150.

.

You can now plug in r and theta into the standard polar form for a number:

### Example Question #3 : Express Complex Numbers In Polar Form

Express the complex number in polar form.

**Possible Answers:**

**Correct answer:**

The figure below shows a complex number plotted on the complex plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

The polar form of a complex number is . We want to find the real and complex components in terms of * *and where * *is the length of the vector and is the angle made with the real axis.

We use the Pythagorean Theorem to find :

We find by solving the trigonometric ratio

Using ,

Then we plug and into our polar equation to obtain

### Example Question #4 : Express Complex Numbers In Polar Form

What is the polar form of the complex number ?

**Possible Answers:**

**Correct answer:**

The correct answer is

The polar form of a complex number is where is the modulus of the complex number and is the angle in radians between the real axis and the line that passes through ( and ). We can solve for and easily for the complex number :

which gives us

### Example Question #1 : Express Complex Numbers In Polar Form

Express the complex number in polar form:

**Possible Answers:**

**Correct answer:**

Remember that the standard form of a complex number is: , which can be rewritten in polar form as: .

To find r, we must find the length of the line by using the Pythagorean theorem:

To find , we can use the equation

Note that this value is in radians, NOT degrees.

Thus, the polar form of this equation can be written as

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