Question

Represent the complex number 3 − i graphically and write it in trigonometric form.

Answer 1

To graph 3 - i, simply graph the point (3,-1) but do so on the complex plane. Remember the complex plane is a cartesian plane with the x axis as the real number line and the y axis as the imaginary number line

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To convert to trignometric form r*( cos(theta) + i*sin(theta) ), use the formulas


theta = arctan(y/x)

and

r = sqrt(x^2 + y^2)

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theta = arctan(y/x)


theta = arctan(-1/3)


theta = -18.4349


r = sqrt(x^2 + y^2)


r = sqrt(1^2 + (-3)^2)


r = sqrt(10)


So the number in trigonometric form is


sqrt(10)*( cos(-18.4349) + i*sin(-18.4349) )

Answer 2

√10(cos 208◦+ i sin 208◦) ?

Answer 3

√10(cos 342◦+ i sin 342◦) ??

Answer 4

√10(cos 137◦+ i sin 137◦) ???

Answer 5

Well remember that cos(theta) = cos(theta+360)


This means that there are infinitely many choices for theta


But we're in luck since we only have 4 choices


Notice that -18.4349 + 360 = 341.5651 which rounds to 342 (basically 342 is very close to the coterminal of -18.4349)


So the answer is √10(cos 342◦+ i sin 342◦)