Problem 31 - Contemporary Precalculus

Question

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$1.5\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

**Question**: Convert the complex number in its polar form, \(1.5(\cos{\frac{\pi}{6}}+i\sin{\frac{\pi}{6}})\), into its standard rectangular form. **Answer**: The complex number in its standard rectangular form is \(\frac{3\sqrt{3}}{4} + \frac{3}{4}i\).

Step 1: Find the value of \(a\)

To find the value of \(a\), we multiply the magnitude \(r\) by the cosine of the argument \(\theta\). In this case, we have: $$ a = 1.5 \cos \frac{\pi}{6} $$ Recall that \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), so we can simplify our expression for \(a\) to: $$ a = 1.5 \cdot \frac{\sqrt{3}}{2} $$ Now, multiply 1.5 by \(\frac{\sqrt{3}}{2}\) to get the final value of \(a\): $$ a = \frac{3\sqrt{3}}{4} $$

Step 2: Find the value of \(b\)

To find the value of \(b\), we multiply the magnitude \(r\) by the sine of the argument \(\theta\). In this case, we have: $$ b = 1.5 \sin \frac{\pi}{6} $$ Recall that \(\sin \frac{\pi}{6} = \frac{1}{2}\), so we can simplify our expression for \(b\) to: $$ b = 1.5 \cdot \frac{1}{2} $$ Now, multiply 1.5 by \(\frac{1}{2}\) to get the final value of \(b\): $$ b = \frac{3}{4} $$

Step 3: Write the complex number in standard form

Now that we have the values of \(a\) and \(b\), we can write the complex number in the standard rectangular form: $$ a + bi = \frac{3\sqrt{3}}{4} + \frac{3}{4}i $$

Key Concepts

Converting Polar to Rectangular Form

Understanding how to convert complex numbers from polar to rectangular form is crucial for comprehending how these numbers operate within a two-dimensional plane. A complex number in polar form can be expressed as \(r(cos\theta + i\sin\theta)\), where \(r\) is the magnitude and \(\theta\) the argument of the number. To convert this into rectangular form, denoted as \(a+bi\), we use trigonometry. Specifically, we find \(a\) by multiplying \(r\) with \(\cos(\theta)\) and \(b\) by multiplying \(r\) with \(\sin(\theta)\).

For instance, if we start with the complex number \(1.5(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6})\), we calculate \(a\) and \(b\) by using special trigonometric values known for \(\frac{\pi}{6}\). With these values found, we just place \(a\) and \(b\) into the rectangular form \(a+bi\) to finalize the conversion.

Cosine and Sine Values of Special Angles

Cosine and sine functions are fundamental in trigonometry, and their values at special angles are often memorized to facilitate calculations involving triangles and circles. Special angles include 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) radians, which correspond to 0, 30, 45, 60, and 90 degrees, respectively.

For example, as seen in our exercise, \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin\frac{\pi}{6} = \frac{1}{2}\). Knowing these values, you can swiftly compute the real and imaginary parts of a complex number when converting from polar to rectangular form without having to refer to a calculator or a trigonometric table.

Multiplying Complex Numbers

When it comes to operations with complex numbers, multiplication might look daunting but follows clear rules. To multiply complex numbers, you typically multiply them as if you're dealing with binomials. That is, for two complex numbers \(a+bi\) and \(c+di\), their product is \(ac + adi + bci + bdi^2\). Given that \(i^2 = -1\), you can simplify the expression to \(ac - bd + (ad + bc)i\).

Multiplication of complex numbers can also be done easily in polar form by simply multiplying the magnitudes and adding the arguments of the complex numbers. This operation showcases the beauty and symmetry of complex numbers, as it ties closely with rotation and scaling in the complex plane.

Rectangular Form of Complex Numbers

The rectangular form of a complex number is the most familiar way to express these numbers. It's written as \(a+bi\), where \(a\) is the real part and \(b\) is the imaginary part of the complex number. This form is highly intuitive as it allows one to easily visualize the complex number as a point in the complex plane, with \(a\) indicating horizontal displacement (along the x-axis) and \(b\) indicating vertical displacement (along the y-axis).

Using algebraic operations on complex numbers in rectangular form is straightforward. Addition and subtraction involve combining like terms, while multiplication and division require a few more steps but still utilize fundamental algebraic principles. The rectangular form is particularly useful in engineering and physics, where complex numbers often represent physical quantities like electrical impedance or wave functions.