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Chapter 2: Q9P (page 69)

Find each of the following in rectangular form x + iyand check your results by computer. Remember to save time by doing as much as you can in your head.
$s i n (蟺 - i In 3)$ .

Short Answer

Expert verified

The rectangular form of the given question is, sin $(蟺 - i In 3) = \frac{4 i}{3}$ .

Step by step solution

01

Given Information.

The given expression is $\sin (蟺 - i In 3)$ .

02

Meaning of rectangular form.

Representing the complex number in rectangular form means writing the given complex number in the form of x = iy, in which is the real part and y is the imaginary part.

03

Step 3: Put the value in the formula.

Use the complex formula $\sin 胃 = \frac{e^{i 胃} - e^{- i 胃}}{2 i}$ to rewrite the above expression.

And $\sin (蟺 - i In 3)$ can written as,

$\sin (蟺 - i In 3) = \sin 蟺 . \cos (i In 3) - \cos 蟺 (i In 3) \sin (蟺 - i In 3) = \sin (i In 3)$

Now,

$\sin (i In 3) = \frac{e^{(i In 3) i} - e^{- (i In 3) i}}{2 i} = \frac{e^{(- In 3)} - e^{(In 3)}}{2 i} = \frac{e^{(In 3^{- 1})} - e^{(In 3)}}{2 i}$

$= \frac{\frac{1}{3} - 3}{2 i}$

$\sin (i In 3) = - \frac{8}{2 i 脳 3} \sin (i In 3) = - \frac{4}{3 i} \sin (i In 3) = - \frac{4 i}{3}$

So,

$\sin (蟺 - i In 3) = \sin (i In 3)$

Substitute the value of $\sin (蟺 - i In 3)$ in the above equation.

$\sin (蟺 - i In 3) = \frac{4 i}{3}$

Therefore, the rectangular form of $\sin (蟺 - i In 3)$ is $\frac{4 i}{3}$ .

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Most popular questions from this chapter

Describe geometrically the set of points in the complex plane satisfying the following equations.

$| z + 3 i | = 4$ .

Question: Express the following complex numbers in the form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others鈥攖ry it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

25. ${[\frac{1 \sqrt{2}}{1 + i}]}^{12}$

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.

17-12i.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$R e (z^{2}) = 4$ .

Solve for all possible values of the real numbers $x$ and $y$ in the following equations.

$x + i y = y + i x$

See all solutions

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