MATH152 April 2013
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Question A 14
Let . Express the complex number in the form where a and b are real numbers.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Use Euler's formula, which states that
and find the value of x that gives the given complex number. then take powers.
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Euler's formula states that for a complex number given in argument form, we have
In the case of the complex number
we can either see that this corresponds to the point at distance 1 of the origin on the diagonal in the first quadrant of the complex plan and hence has an argument of 45° or of π/4; or you might directly remember the values of the sine and cosine of π/4. Either way, we obtain that
Taking the 2010 of is much easier in this shape. We have
In this notation we see that the argument of is which remains the same if we remove or add , so since we have that the angle and so
Using Euler's formula again yields
Using Euler's formula the other way around we obtain that
Since and .
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