Polar Form
Convert the following into polar form:
Expand Hint
Complex numbers can be written in both rectangular and polar form. For rectangular form, a complex number is represented by both its real and imaginary components:
$$$z=a+jb$$$
where
$$a$$
is the real component,
$$b$$
is the imaginary component, and
$$j=\sqrt{-1}$$
which is often expressed as
$$i=\sqrt{-1}$$
.
Hint 2
Polar form is
$$$z=c\: \angle \: \theta $$$
where
$$c=\sqrt{a^2+b^2}$$
(the vector’s length) and
$$\theta=tan^{-1}(b/a)$$
. Note:
$$a=c\:cos(\theta)$$
and
$$b=c\:sin(\theta)$$
.
Complex numbers can be written in both rectangular and polar form. For rectangular form, a complex number is represented by both its real and imaginary components:
$$$z=a+jb$$$
where
$$a$$
is the real component,
$$b$$
is the imaginary component, and
$$j=\sqrt{-1}$$
which is often expressed as
$$i=\sqrt{-1}$$
.
Thus, the rectangular coordinate form can be rewritten as:
$$$7+\sqrt{-1}\sqrt{16}\to 7+4i$$$
where
$$a=7$$
and
$$b=4$$
.
Polar form is:
$$$z=c\: \angle \: \theta $$$
where
$$c=\sqrt{a^2+b^2}$$
(the vector’s length) and
$$\theta=tan^{-1}(b/a)$$
. Note:
$$a=c\:cos(\theta)$$
and
$$b=c\:sin(\theta)$$
.
Therefore,
$$$c=\sqrt{7^2+4^2}=\sqrt{49+16}=\sqrt{65}=8.06$$$
$$$\theta=tan^{-1}(\frac{4}{7})=29.7^{\circ}$$$
Finally, the polar form is:
$$$8.06 \: \angle \: 29.7^{\circ}$$$
$$$8.06 \: \angle \: 29.7^{\circ}$$$
Time Analysis
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Similar Problems from FE Section: Algebra of Complex Numbers
345. Binomial Product
552. Polar
611. The Polar Form
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