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Simplify four cos of 90 plus π sin of 90 multiplied by five cos of 80 plus π sin 80 multiplied by four cos of 45 plus π sin of 45, giving your answer in trigonometric form.
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Recall the product formula.
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This says that, for two complex numbers expressed in polar form, π one with a modulus of π one and an argument of π one and π two with a modulus of π two and an argument of π two, their product can be found by multiplying the moduli and adding the arguments.
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We can extend this to three complex numbers and find the product of the three complex numbers that weβve been given.
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Letβs begin by multiplying their moduli.
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Thatβs four, five, and four, which is 80.
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Next, weβll add their arguments.
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Thatβs 90, 80, and 45, which is 215.
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In trigonometric or polar form then, the product of these three complex numbers is 80 multiplied by cos of 215 degrees plus π sin of 215.