Question

Write each complex number in rectangular form. $$ 3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right) $$

Short answer

-3i

Step-by-step solution

- Identify the angle

The given complex number is in polar form: \[3\big(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}\big).\]Here, the angle is \( \frac{3\pi}{2} \).

- Evaluate \(\cos \frac{3\pi}{2}\)

\(\cos \frac{3\pi}{2} \) is evaluated as follows:\[\cos \frac{3\pi}{2} = 0.\]

- Evaluate \(\sin \frac{3\pi}{2}\)

\(\sin \frac{3\pi}{2} \) is evaluated as follows:\[\sin \frac{3\pi}{2} = -1.\]

- Substitute the values

Substitute the evaluated values of \(\cos \frac{3\pi}{2}\) and \(\sin \frac{3\pi}{2}\) into the polar form:\[3 \big(0 + i \cdot (-1)\big).\]

- Simplify the expression

Simplify the expression to get the rectangular form:\[3 \big(0 + i \cdot (-1)\big) = 3 \cdot 0 + 3i \cdot (-1) = 0 - 3i.\]Thus, the rectangular form is \(-3i\).

Key concepts

rectangular form

A complex number in rectangular form is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit. This form makes it easy to identify the real and imaginary parts of a complex number.
To simplify complex numbers, we often perform arithmetic operations on the real and imaginary parts separately. For example, if you have \(3 + 4i\) and \(2 + 5i\), you can add them by adding the real parts together and the imaginary parts together: \((3 + 2) + (4i + 5i) = 5 + 9i\).
In some cases, you might be given a complex number in another form, like polar form, and asked to convert it to rectangular form. We'll explore this in the next few sections.

polar form

The polar form of a complex number is given by \(r(\text{cos}\theta + i\text{sin}\theta)\), where \(r\) is the magnitude (or modulus) and \(\theta\) is the angle (or argument). This form uses trigonometric functions to represent the complex number based on its position in the polar coordinate system.
To convert a complex number from polar to rectangular form, you need to evaluate the cosine and sine functions at the given angle \(\theta\), and then multiply these values by the magnitude \(r\).
For example, given \(3(\text{cos} \frac{3\theta}{2} + i\text{sin} \frac{3\theta}{2})\), you identify the angle \(\frac{3\theta}{2}\), evaluate the trigonometric functions at this angle, and then simplify the expression. This is how you would convert the given example into rectangular form.

trigonometric functions

Trigonometric functions, specifically sine and cosine, are fundamental to working with complex numbers in polar form. The cosine of an angle \(\theta\), represented as \(\text{cos}\theta\), gives the horizontal coordinate on the unit circle, while the sine of the angle \(\text{sin}\theta\) gives the vertical coordinate.
When converting from polar to rectangular form, you substitute the values of \(\text{cos}\theta\) and \(\text{sin}\theta\) into the formula \(r(\text{cos}\theta + i\text{sin}\theta)\). For instance, for \(\theta = \frac{3\theta}{2}\), you calculate \(\text{cos} \frac{3\theta}{2} = 0\) and \(\text{sin} \frac{3\theta}{2} = -1\), and then you multiply these by the magnitude \(r\) to find the rectangular coordinates.
Understanding these trigonometric functions is key to mastering conversions and fully grasping the interplay between rectangular and polar representations of complex numbers.