De Moivre's Theorem: Positive Powers

Error

There was an error loading the page.

Contributors

Newcastle University Mathematics and Statistics

Feedback

From users who are not members of Content created by Newcastle University :

Anum Khalid

said

Ready to use

2 years ago

History

○

Anum Khalid 2 years ago

Gave some feedback: Ready to use

○

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Created this.

Name	Status	Author	Last Modified
De Moivre's Theorem: Positive Powers	Ready to use	Newcastle University Mathematics and Statistics	07/04/2023 08:05
Maria's copy of De Moivre's Theorem: Positive Powers	draft	Maria Aneiros	25/05/2019 08:24
Shaheen's copy of De Moivre's Theorem: Positive Powers	Ready to use	Shaheen Charlwood	24/02/2020 13:39

There are 25 other versions that do you not have access to.

Question statement

Use de Moivre's theorem to write the following complex numbers in the form $a+bi$ .

Note that for these questions, arguments of complex numbers lie in the range $-\pi \lt \theta \le \pi$ .

Important: When calculating the final answer in part (iii) of each question, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.

Give any introductory information the student needs.

			Name	Type	Generated Value

co	integer	3
ans1	number	1.414
arg2	number	-0.322
ans3	number	11.314
ans4	number	22.627
tb4	number	-7696
b4	number	-7696.000
b1	integer	-1
b2	integer	-1
b3	number	-16.000
d4	integer	-4
d2	integer	8
z6	number	9 - 4i
q1	string	The complex number is in the f
q3	string	The complex number is in the t
q2	string	The complex number is in the s
q4	string	The complex number is in the f
s3	integer	-1
s2	integer	-1
s1	integer	-1
s7	integer	1
s6	integer	1
s5	integer	-1
s4	integer	1
m4	string	The complex number is in the s
m1	string	The complex number is in the t
m3	string	The complex number is in the f
arg1	number	-2.356
z3	number	8 + 8i
tb3	number	-16
ta4	number	-30672
arg3	number	0.785
tol	number	0.001
ta3	number	-16
arg4	number	-2.356
m2	string	The complex number is in the f
a1	integer	-1
a3	number	-16.000
s8	integer	1
a4	number	-30672.000
z4	number	-16 - 16i
z5	number	-30672 - 7696i
ans2	number	3.162
z1	number	-1 - i
z2	number	3 - i
c4	integer	9
f	integer	9
a2	integer	3
t	integer	3
n2	integer	9
n4	integer	9
c2	integer	8

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
s4*random(1..3)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

← Depends on:

→ Used by:

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Mathematical expression
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Find the modulus and argument of $\var{z1}$ to 3 decimal places.

(i) $|\var{z1}|\;=\;$ Gap 0., to 3 decimal places.

(ii) $\arg(\var{z1})\;=\;$ Gap 1. radians, to 3 decimal places.

Hence find:

(iii) $(\var{z1})^{\var{n2}}\;=\;$ Gap 2.

Input as a complex number, with real and imaginary parts integral values.

Advice

Given a complex number $z=r(\cos(\theta)+i\sin(\theta))$ de Moivre's theorem states that $z^n=r^n(\cos(n\theta)+i\sin(n\theta))$ for an integer power $n$ .
So if we know the modulus $r$ and the argument $\theta$ for $z$ then the theorem provides a way of calculating $z^n$ .

As usual, you must be careful that the argument is calculated correctly by paying attention to the quadrant of the complex plane in which the complex number lies.

Also remember that for this question, arguments of complex numbers lie in the range $-\pi \lt \theta \le \pi$ .

With the above in mind we can now answer the questions:

Modulus

$\begin{eqnarray*} |\var{z1}|&=&\sqrt{(\var{a1})^2+(\var{b1})^2}\\ &=& \var{abs(z1)}\\ &=&\var{ans1} \end{eqnarray*}$ to 3 decimal places.

Note that $r^{\var{n2}}=|(\var{z1})^{\var{n2}}| =\var{abs(z1)}^{\var{n2}}=\var{abs(z1)^n2}$ which we will use in the calculation for $(\var{z1})^{\var{n2}}$

Argument

{m1}.
Hence we see that:

$\begin{eqnarray*} \arg(\var{z1}) &=& \var{arg(z1)}\\ &=& \var{arg1}\; \mbox{radians} \end{eqnarray*}$ to 3 decimal places.

We have $\arg((\var{z1})^{\var{n2}})=\var{n2}\times \var{arg(z1)} = \var{n2*arg(z1)}$ radians.

Hence we have

$\begin{eqnarray*}(\var{z1})^{\var{n2}} &=& \var{abs(z1)^n2}(\cos(\var{n2*arg(z1)})+\sin(\var{n2*arg(z1)})i)\\ &=& \var{abs(z1)^n2}\cos(\var{n2*arg(z1)})+\var{abs(z1)^n2}\times\sin(\var{n2*arg(z1)})i\\ &=& \simplify[std]{{a3}+{b3}i}. \end{eqnarray*}$

Modulus

$\begin{eqnarray*} |\var{z2}|&=&\sqrt{(\var{a2})^2+(\var{b2})^2}\\ &=& \var{abs(z2)}\\ &=&\var{ans2} \end{eqnarray*}$ to 3 decimal places.

Note that $r^{\var{n4}}=|(\var{z2})^{\var{n4}}| =\var{abs(z2)}^{\var{n4}}=\var{abs(z2)^n4}$ which we will use in the calculation for $(\var{z2})^{\var{n4}}$

Argument

{m2}.
Hence we see that:

$\begin{eqnarray*} \arg(\var{z2}) &=& \var{arg(z2)}\\ &=& \var{arg2}\; \mbox{radians} \end{eqnarray*}$ to 3 decimal places.

We have $\arg((\var{z2})^{\var{n4}})=\var{n4}\times \var{arg(z2)} = \var{n4*arg(z2)}$ radians.

Hence we have

$\begin{eqnarray*}(\var{z2})^{\var{n4}} &=& \var{abs(z2)^n4}(\cos(\var{n4*arg(z2)})+\sin(\var{n4*arg(z2)})i)\\ &=& \var{abs(z2)^n4}\cos(\var{n4*arg(z2)})+\var{abs(z2)^n4}\times\sin(\var{n4*arg(z2)})i\\ &=& \simplify[std]{{a4}+{b4}i}. \end{eqnarray*}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exams:

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Anum Khalid 2 years ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Modulus

Argument

Modulus

Argument

Loading...

Error

De Moivre's Theorem: Positive Powers

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Anum Khalid 2 years ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Modulus

Argument

Modulus

Argument

Generated value: `integer`