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Chapter 7: Problem 17

Express each complex number in trigonometric form. $$1-\sqrt{3} i$$

Short Answer

Expert verified
The complex number \(1-\sqrt{3}i\) in trigonometric form is \(2 \cdot (\cos{5蟺/3} + i\sin{5蟺/3})\).

Step by step solution

01

Calculate r

We calculate 'r' using the formula \(r=\sqrt{a^2 + b^2} => r= \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1+3} = 2\)
02

Calculate 胃

We calculate '胃' using the formula \(胃 = \arctan{\frac{-\sqrt{3}}{1}} = \arctan{-\sqrt{3}}\). But, we know from the unit circle value, that \(arctan{-\sqrt{3}}\) gives us -蟺/3. By considering the quadrant that the complex number is in (since 'a' is positive and 'b' is negative , the complex number lies in the 4th quadrant), we adjust the value of 胃 by adding 2蟺 , hence \(胃 = -蟺/3 + 2蟺 = 5蟺/3\)
03

Express in Trigonometric Form

The complex number in trigonometric form will be \(r \cdot (\cos{\theta} + i\sin{\theta}) \). By substituting the values we got, the final answer will be \(2 \cdot (\cos{5蟺/3} + i\sin{5蟺/3})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Form of Complex Numbers
Complex numbers can be portrayed in a trigonometric form, which offers a different perspective than the standard rectangular form. While complex numbers have a real and an imaginary part, the trigonometric form represents them using a radius and an angle.

The formula for this trigonometric form is given by:
  • Radius, \(r\), is calculated as \(\sqrt{a^2 + b^2}\) where \(a\) and \(b\) are the real and imaginary parts respectively.
  • Angle, \(\theta\), is determined using the arctangent function \(\theta = \arctan\left(\frac{b}{a}\right)\).
  • Thus, the complex number can be expressed as \(r(\cos{\theta} + i\sin{\theta})\).
Understanding the trigonometric form is incredibly useful for multiplication and division of complex numbers, as it simplifies these processes significantly.
Understanding the Unit Circle
The unit circle is a critical tool in understanding angles in trigonometry, especially when dealing with complex numbers. It is a circle with a radius of 1, centered at the origin of the coordinate plane.

The unit circle helps us find the values of sine and cosine for any angle, which are key components of the trigonometric form.
  • Angles in the unit circle are typically measured in radians rather than degrees.
  • For example, an angle of \(-\pi/3\) or \(5\pi/3\) points to the same location on the unit circle, corresponding to different rotations.
  • The positive direction is counterclockwise, whereas a clockwise rotation indicates a negative angle.
By leveraging the unit circle, we can easily interpret the angle associated with a complex number based on its real and imaginary components.
The Arctan Function and Its Role
The arctan function, short for arctangent, is essential for finding the angle in the trigonometric form of complex numbers. It is the inverse of the tangent function and helps determine the angle \(\theta\) given the ratio of the opposite side to the adjacent side (\(\frac{b}{a}\)).

However, using \(\arctan\) requires careful quadrant consideration since it naturally outputs angles between \(-\pi/2\) and \(\pi/2\).
  • In our example, when finding \(\theta\) using \(\arctan(-\sqrt{3})\), we initially get \(-\pi/3\).
  • This angle needed to be adjusted to \(5\pi/3\) because the complex number resides in the fourth quadrant, where the real part is positive and the imaginary part is negative.
  • Adjustments to \(\theta\) ensure correct angle representation, based on the complex number's quadrant placement.
Remember that such adjustments are typical when translating between arctan outputs and complex number quadrant placement.

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Most popular questions from this chapter

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