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Given that π§ one is equal to two cos of five π minus two π plus π sin of five π minus two π and π§ two equals four cos of four π minus three π plus π sin of four π minus three π, find π§ one multiplied by π§ two.
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We have been given two complex numbers represented in polar or trigonometric form.
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And weβre looking to find their product.
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Remember, to multiply complex numbers in polar form, we multiply their moduli.
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And we add their arguments.
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And the general form of a complex number in polar form is π cos π plus π sin π, where π is the modulus and π is the argument.
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Weβll compare this general form to the complex numbers in our question.
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The modulus of our first complex number π§ one is two.
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And the modulus of our second complex number is four.
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The argument of our first complex number is five π minus two π.
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And the argument of our second complex number is four π minus three π.
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We said that, to find the modulus of the product of these two complex numbers, we need to find the product of their moduli.
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Thatβs two multiplied by four, which is of course eight.
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And we said that, to find the argument of π§ one multiplied by π§ two, we add their respective arguments.
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Thatβs five π minus two π plus four π minus three π.
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We can collect like terms.
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And we see that the argument of the product of π§ one and π§ two is nine π minus five π.
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And all thatβs left is to substitute these values into the general form of a complex number in polar form.
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Itβs eight cos of nine π minus five π plus π sin of nine π minus five π.