Answered: 3. 8 + 6j

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3. 8 + 6j

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.5: Trigonometric Form For Complex Numbers

Problem 38E

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Question

Represent each complex number graphically and give the polar form of each

Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.

Expert Solution

Step 1

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(3) The given complex number is $8 + 6 j$ .

Note that $8 + 6 j$ is of the form x+yj where $x = 8, y = 6$ .

Substitute $x = 8, y = 6$ in $r = \sqrt{x^{2} + y^{2}}$ :

$\begin{array}{rcl} r & = & \sqrt{8^{2} + 6^{2}} \\ = & \sqrt{64 + 36} \\ = & \sqrt{100} \\ = & 10 \end{array}$

Step 2

Substitute $x = 8, y = 6$ in $θ = \tan^{- 1} (\frac{y}{x})$ :

$\begin{array}{rcl} θ & = & \tan^{- 1} (\frac{6}{8}) \\ = & 36.87 ° \end{array}$

Since $x = 8, y = 6$ , it is in first quadrant.

Substitute $x = 8, y = 6$ , $θ = 36.87 °, r = 10$ in $x + y j = r (\cos θ + j \sin θ)$ to obtain the polar form:

$8 + 6 j = 10 (\cos (36.87 °) + j \sin (36.87 °))$

Thus, the polar form of $8 + 6 j$ is $10 (\cos (36.87 °) + j \sin (36.87 °))$ .

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