Convert all complex numbers to trigonometric form and then simplify the expression? Write the answer in standard form.
#((2 + 2i)^5(-3+i)^3)/((sqrt3 +i)^10#
2 Answers
Explanation:
As anyone who reads my answers may have noticed, my pet peeve is every trig problem involves a 30/60/90 or 45/45/90 triangle. This one has both, but
I'm going to go out on a limb and guess the question in the book actually read:
Use trigonometric form to simplify
because this way would only involve the Two Tired Triangles of Trig.
Let's convert to trigonometric form, which is just polar form written
Then by De Moivre's Thorem
Let's convert each factor.
That's the 30/60/90 triangle in the second quadrant whose cosine is bigger (in magnitude) than its sine.
Similarly,
Now,
I don't want to deal with the trig functions of
Explanation:
In another answer to this question I guessed there was a typo in this question and that
I won't repeat how we determined
But now we have to convert
We're in the second quadrant and the principal value of the inverse tangent is the fourth quadrant.
De Moivre doesn't work very well on a form like this, we get
But we're not stuck. Since the exponent is only
By De Moivre,
We know
That seems like much more work than just cubing
OK, let's do the problem:
Ugh, it never ends. We get