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Chapter 6: Problem 37

In Exercises \(33-42,\) find the standard form of the complex number. Then represent the complex number graphically. $$ \frac{9}{4}\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right) $$

Short Answer

Expert verified
The standard form of the complex number is \(-\dfrac{9\sqrt{2}}{8} + i\ \dfrac{9\sqrt{2}}{8}\), and its graphical representation is a point at (-1.27, 1.27) on the complex plane.

Step by step solution

01

Identify the magnitude and the angle

The magnitude, denoted by \(r\), is \(\dfrac{9}{4}\), and the angle 胃, both in radians, is \(\dfrac{3\pi}{4}\).
02

Apply trigonometric identities

Now, using the identities of the cosine and sine, we calculate the real part \(a = r\cos(胃) = \dfrac{9}{4}\cos\(\dfrac{3\pi}{4}\) = -\dfrac{9\sqrt{2}}{8}\) and the imaginary part \(b = r\sin(胃) = \dfrac{9}{4}\sin\(\dfrac{3\pi}{4}\) = \dfrac{9\sqrt{2}}{8}\).
03

Write in standard form

The standard form of a complex number is \(a + bi\). Substituting the calculated values, we get \(-\dfrac{9\sqrt{2}}{8} + i\ \dfrac{9\sqrt{2}}{8}\).
04

Graphical representation

On the complex plane, plot a point at \((-1.27, 1.27)\) which are the calculated real part (x-coordinate) and the imaginary part (y-coordinate).

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Most popular questions from this chapter

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(-2i(1\ +\ i)\)

In Exercises 99-106, use the formula on page 474 to find all the solutions of the equation and represent the solutions graphically. \(x^5 + 243 = 0\)

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(1 - \sqrt{3}i\)

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(-2(1 + \sqrt{3}i)\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{5(\cos\ 4.3 + i\ \sin\ 4.3)}{4(\cos\ 2.1 + i\ \sin\ 2.1)}\)
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