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bija089 [108]
3 years ago
11

Select <, >, or=. 16 lb ? 240 oz

Mathematics
1 answer:
s344n2d4d5 [400]3 years ago
5 0
16lb is greater than 240oz. if it were to be equal the amount of ounces would be 256oz.
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Use De Moivre's theorem to write the complex number in trigonometric form: (cos(3pi/5) + i sin (3pi/5))^3
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By De Moivre's theorem,

\left(\cos\dfrac{3\pi}5+i\sin\dfrac{3\pi}5\right)^3=\boxed{\cos\dfrac{9\pi}5+i\sin\dfrac{9\pi}5}

We can stop here ...

# # #

... but we can also express these trig ratios in terms of square roots. Let x=\dfrac\pi5 and let c=\cos x. Then recall that

\cos5x=c^5-10c^3\sin^2x+5c\sin^4x

\cos5x=c^5-10c^3(1-c^2)+5c(1-2c^2+c^4)

\cos5x=16c^5-20c^3+5c

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4c^2-2c-1=0\implies c=\dfrac{1+\sqrt5}4

because we know to expect \cos x>0. Then from the Pythagorean identity, and knowing to expect \sin x>0, we get

\sin x=\sqrt{1-c^2}=\sqrt{\dfrac58-\dfrac{\sqrt5}8}

Both \cos and \sin are 2\pi-periodic, so that

\cos\dfrac{9\pi}5=\cos\left(\dfrac{9\pi}5-2\pi\right)=\cos\left(-\dfrac\pi5\right)=\cos\dfrac\pi5

and

\sin\dfrac{9\pi}5=\sin\left(-\dfrac\pi5\right)=-\sin\dfrac\pi5

so that the answer we left in trigonometric form above is equal to

\cos\dfrac\pi5-i\sin\dfrac\pi5=\boxed{\dfrac{1+\sqrt5}4+i\sqrt{\dfrac58+\dfrac{\sqrt5}8}}

7 0
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