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Chapter 6: Problem 30

Write each complex number in rectangular form. If necessary, round to the nearest tenth. $$10\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)$$

Short Answer

Expert verified
The rectangular form of the given complex number is \( -10 - 10\sqrt{3}i \).

Step by step solution

01

Determine r and 胃

First, identify \( r \) and \( \theta \) from the polar form of the complex number. In the given complex number \( 10(\cos 210^{\circ}+i \sin 210^{\circ}) \), \( r = 10 \) and \( \theta = 210掳 \).
02

Conversion to Radians

Convert the angle from degrees to radians. Note that \( 1掳 = \frac{\pi}{180} \) radians. So, \( \theta = 210掳 = 210 * \frac{\pi}{180} = \frac{7}{6}\pi \) radians.
03

Compute Real part

Compute the real part (a) of the complex number in rectangular form. It's given by \( a = r \cos\theta \). Therefore, \( a = 10 \cos(\frac{7}{6}\pi) = -10 \).
04

Compute Imaginary part

Determine the imaginary part (b) of the complex number in rectangular form. It's given by \( b = r \sin\theta \). Therefore, \( b = 10 \sin(\frac{7}{6}\pi) = -10\sqrt{3} \).
05

Build the Rectangular Complex Number

Express the complex number in rectangular form using the above computed values. Therefore, the rectangular form is \( a + bi = -10 - 10\sqrt{3}i \).

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Most popular questions from this chapter

Solve: \(\cos 2 x-\sin x=0,0 \leq x<2 \pi\) (Section \(5.5, \text { Example } 8)\)

A force of 80 pounds on a rope is used to pull a box up a ramp inclined at \(10^{\circ}\) from the horizontal. The rope forms an angle of \(33^{\circ}\) with the horizontal. How much work is done pulling the box 25 feet along the ramp?

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}+9 \mathbf{j}$$

Describe a test for symmetry with respect to the line \(\theta=\frac{\pi}{2}\) in which \(r\) is not replaced.

Exercises \(81-83\) will help you prepare for the material covered in the next section. Two airplanes leave an airport at the same time on different runways. The first plane, flying on a bearing of \(\mathrm{N} 66^{\circ} \mathrm{W},\) travels 650 miles after two hours. The second plane, flying on a bearing of \(\mathrm{S} 26^{\circ} \mathrm{W},\) travels 600 miles after two hours. Illustrate the situation with an oblique triangle that shows how far apart the airplanes will be after two hours.
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