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3 years ago

PLEASE HELP!!

Mathematics

1 answer:

Alinara [238K]3 years ago

8 0

Point B (2, -3)
Point C (2, 3)

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Find e^cos(2+3i) as a complex number expressed in Cartesian form.

ozzi

Answer:

The complex number $e^{\cos(2+31)} = \exp(\cos(2+3i))$ has Cartesian form

$\exp\left(\cosh 3\cos 2\right)\cos(\sinh 3\sin 2)-i\exp\left(\cosh 3\cos 2\right)\sin(\sinh 3\sin 2)$ .

Step-by-step explanation:

First, we need to recall the definition of $\cos z$ when $z$ is a complex number:

$\cos z = \cos(x+iy) = \frac{e^{iz}+e^{-iz}}{2}$ .

Then,

$\cos(2+3i) = \frac{e^{i(2+31)} + e^{-i(2+31)}}{2} = \frac{e^{2i-3}+e^{-2i+3}}{2}$ . (I)

Now, recall the definition of the complex exponential:

$e^{z}=e^{x+iy} = e^x(\cos y +i\sin y)$ .

So,

$e^{2i-3} = e^{-3}(\cos 2+i\sin 2)$

$e^{-2i+3} = e^{3}(\cos 2-i\sin 2)$ (we use that $\sin(-y)=-\sin y)$ .

Thus,

$e^{2i-3}+e^{-2i+3} = e^{-3}\cos 2+ie^{-3}\sin 2 + e^{3}\cos 2-ie^{3}\sin 2)$

Now we group conveniently in the above expression:

$e^{2i-3}+e^{-2i+3} = (e^{-3}+e^{3})\cos 2 + i(e^{-3}-e^{3})\sin 2$ .

Now, substituting this equality in (I) we get

$\cos(2+3i) = \frac{e^{-3}+e^{3}}{2}\cos 2 -i\frac{e^{3}-e^{-3}}{2}\sin 2 = \cosh 3\cos 2-i\sinh 3\sin 2$ .

Thus,

$\exp\left(\cos(2+3i)\right) = \exp\left(\cosh 3\cos 2-i\sinh 3\sin 2\right)$

$\exp\left(\cos(2+3i)\right) = \exp\left(\cosh 3\cos 2\right)\left[ \cos(\sinh 3\sin 2)-i\sin(\sinh 3\sin 2)\right]$ .

5 0

3 years ago

Sin(-180° + θ) = ________

Svet_ta [14]

Answer:

$\sin\left(-180^o+\theta\right)=-\sin\left(\theta\right)$

Step-by-step explanation:

We need to find the value of $\sin(-180^o+ \theta)$

So let's apply formula

$\sin\left(A+B\right)=\sin\left(A\right)\cos\left(B\right)+\sin\left(B\right)\cos\left(A\right)$

$\sin\left(-180^o+\theta\right)=\sin\left(-180^o\right)\cos\left(\theta\right)+\sin\left(\theta\right)\cos\left(-180^o\right)$

$\sin\left(-180^o+\theta\right)=0\cdot\cos\left(\theta\right)+\sin\left(\theta\right)\cdot\left(-1\right)$

$\sin\left(-180^o+\theta\right)=0-\sin\left(\theta\right)$

$\sin\left(-180^o+\theta\right)=-\sin\left(\theta\right)$

8 0

3 years ago

Help me.. pls pls pls pls

AveGali [126]

The other side is 5 hope this helps

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3 years ago

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Tim made a down payment of $1,000. He pays $450 for 36 months. What is the total amount he pays for the car?

lesya692 [45]

Total amount payable= 1000+450*36
Total amount payable=1000+16200
Total amount payable=<span>$17200</span>

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4 years ago

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Three boxes of cereal weigh 379.4 grams, 424.25 grams and 379.37 grams.What is the difference between the lightest and the heavi

Alisiya [41]

You subtract the heaviest amount which is 424.25 grams in this case from the smallest, 379.37 grams.Your answer should be C.

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3 years ago