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Chapter 2: Q7P (page 52)

Question:First simplify each of the following numbers to the x+iyform or to the $r e^{颈胃}$ form. Then plot the number in the complex plane.
$(\frac{3 + i}{2 + i})$

Short Answer

Expert verified

The complex number can be written as $\sqrt{2} e^{- i 0.141}$ .

The graph is mentioned below:

Step by step solution

01

Given Information

Thecomplex number is $(\frac{3 + i}{2 + i})$ .

02

Definition of the Complex number.

A complex number is a number that has real and imaginary numbers and is written in the form as:

$z = a + i b$ where a and b are real, and z is the complex number.

03

Find the value

Factorize the complex number.

$z = \frac{3 + i}{2 + i} 脳 \frac{2 - i}{2 - i} = \frac{6 - i - i^{2}}{4 - i^{2}} = \frac{7}{5} - \frac{i}{5}$

Find x and y

$x = \frac{7}{5} = - \frac{1}{5}$

Find the value of r .

$r = \sqrt{x^{2} + y^{2}} r = \sqrt{\frac{49}{25} + \frac{1}{25}} = \sqrt{\frac{50}{25}} = \sqrt{2}$

Find the value of .

$胃 = \sin^{- 1} (\frac{y}{r}) 胃 = \sin^{- 1} (\frac{\frac{1}{5}}{\sqrt{2}}) = - 0.141 (\frac{3 + i}{2 + i}) = \sqrt{2} e^{- i 0.141}$

The complex number can be written as $\sqrt{2} e^{- i 0.141}$ .

The graph is mentioned below as:

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Most popular questions from this chapter

Find the power series for $e^{x} c o s x$ and for $e^{x} s i n x$ from the series for $e z$ in the following way: Write the series for $e^{z}$ ; put $z = x + i y$ . Show that $e^{z} = e^{x} (c o s y + i s i n y)$ ; take real and imaginary parts of the equation, and put $y = x$ .

Question: First simplify each of the following numbers to the x+iyform or to the $r e^{i 胃}$ form. Then plot the number in the complex plane.

$\frac{5 - 2 i}{5 + 2 i}$ .

Question: Find vand a if $z = c o s 2 t + i s i n 2 t$ , can you describe the motion.

Question:First simplify each of the following numbers to the x+iy form or to the $r e^{i 胃}$ form. Then plot the number in the complex plane.

1.6-2.7i .

For each of the following numbers, first visualize where it is in the complex plane. With a little practice, you can quickly find $x, y, r, 胃$ in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also, plot the complex conjugate of the number.

$7 (c o s 110^{掳} - i s i n 110^{掳})$ .

See all solutions

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