Without using a calculator, order the following numbers from least to greatest.<br> 5<br> 39<br> 6

Simplify the complex numbers using de Moivre's Theorem and match them with their solutions.

alexandr1967 [171]

For the numbers in $a+bi$ form, convert to polar form:

$1+i=\sqrt2\dfrac{1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)$

By DeMoivre's theorem,

$(1+i)^5=(\sqrt2)^5\left(\cos\dfrac{5\pi}4+i\sin\dfrac{5\pi}4\right)=4\sqrt2\dfrac{-1-i}{\sqrt2}=-4-4i$

$-1+i=\sqrt2\dfrac{-1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4}\right)$

$\implies(-1+i)^6=(\sqrt2)^6\left(\cos\dfrac{18\pi}4+i\sin\dfrac{18\pi}4\right)=8i$

$\sqrt3+i=2\dfrac{\sqrt3+i}2=2\left(\cos\dfrac\pi6+i\sin\dfrac\pi6\right)$

$\implies2(\sqrt3+i)^{10}=2^{11}\left(\cos\dfrac{10\pi}6+i\sin\dfrac{10\pi}6\right)=2^{11}\dfrac{1-i\sqrt3}2=2^{10}(1-i\sqrt3)$

For the numbers already in polar form, DeMoivre's theorem can be applied directly:

$2\left(\cos20^\circ+i\sin20^\circ\right)^3=2\left(\cos60^\circ+i\sin60^\circ\right)=2\dfrac{1+i\sqrt3}2=1+i\sqrt3$

$2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)^4=2(\cos\pi+i\sin\pi)=-2$

At second glance, I think the 2s in the last two numbers should also be getting raised to the 3rd and 4th powers:

$\left(2(\cos20^\circ+i\sin20^\circ)\right)^3=8\left(\cos60^\circ+i\sin60^\circ\right)=4+4\sqrt3$

$\left(2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)\right)^4=16(\cos\pi+i\sin\pi)=-16$

4 0

3 years ago

Michael bought a shirt with 2/5 of his money. Then he bought a jacket that cost $5 more than the shirt. He spent $105 altogether

TEA [102]

Answer:

vb hyjv bnhyn g

Step-by-step explanation:

8 0

3 years ago

Please help stuck on probelem

Maurinko [17]

<h3>Answer: 36 square units</h3>

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Explanation:

Draw a horizontal line to cut the figure into a rectangle on top and a triangle down below.

The rectangle has length and width of 6 and 4 (horizontal and vertical components respectively). The area of this rectangle is 6*4 = 24 square units.

The triangle has a base of 6 and a height of 4. The base and height of any triangle are always perpendicular.

The area of the triangle is base*height/2 = 6*4/2 = 24/2 = 12 square units. If you were to cut out half of the triangle and rearrange things, you'll find that a rectangle can be formed. This rectangle is half in area that of the first rectangle we found. This is why we divide by 2 when finding the area of the triangle.

Once you know the two sub-areas, we add them up to get the overall area: 24+12 = 36 square units.

5 0

3 years ago

Read 2 more answers

How do i solve this equation?

Luden [163]

Answer:

x = 28

Step-by-step explanation:

(3x + 6^o) = 90^o

3x = 84^o

x = 28

7 0

3 years ago

Each point on the coordinate plane has an address called

RideAnS [48]

I do not understand what to do , more explicitly sorry :(

8 0

3 years ago

Without using a calculator, order the following numbers from least to greatest. 5 39 6