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3 years ago

Find the fourth roots of 81(cos 320° + i sin 320° ). Write the answer in trigonometric form.

Mathematics

2 answers:

Anna [14]3 years ago

5 0

Answer with explanation:

The given expression which is in complex form is :

=81 (Cos 320°+Sin 320°)------------------------------------(1)

For, a Complex number in the form of

Z=r [Cos A + i Sin A], Can be written as

$Z=re^{iA}$

We have to find four roots of expression (1).

$Z^4=81 (Cos 320^{\circ}+iSin 320^{\circ})\\\\Z=[81\times (Cos(2k\pi + 320^{\circ})+iSin (2k\pi +320^{\circ})]^{\frac{1}{4}}\\\\Z={3^{{4}\times^{\frac{1}{4}}}\times e^{i(\frac{2k\pi +320^{\circ}}{4})}}} \\\\Z=3e^{i(\frac{k\pi}{2}+ 80^{\circ}})\\\\Z_{0}=3(Cos 80^{\circ}+iSin 80^{\circ})\\\\Z_{1}=3(Cos 170^{\circ}+iSin 170^{\circ})\\\\Z_{2}=3(Cos 260^{\circ}+iSin260^{\circ})\\\\Z_{3}=3(Cos 350^{\circ}+iSin 350^{\circ})$

The four values are obtained for, k=0,1,2,3,.

Liono4ka [1.6K]3 years ago

3 0

Using the De Moivre's Theorem, let us work out for the fourth roots of 81(cos 320° + i sin 320° ).

zⁿ = rⁿ (cos nθ + i sin nθ)
z⁴ = 81(cos 320° + i sin 320° )
z = ∜[81(cos 320° + i sin 320° )]
= ∜[3^4 (cos 4*80° + i sin 4*80°)]
= 3(cos 80° + i sin 80°)

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3 years ago

Read 2 more answers

Can anyone help me?

ArbitrLikvidat [17]

Answer:

The factored form of 4m³ – 28m² – 120m is 4m(m – 10)(m + 3). The zeroes of the function would be m = 0, m = –3, and m = 10.

Step-by-step explanation:

I'll give this a shot.

4m³ – 28m² – 120m = 0 — The original expression

4m³ – 28m² – 120m — Did that 0 have any purpose? I just deleted it.

4(m³ – 7m² – 30m) — There's a common factor in here, 4. Let's pull that aside.

4m(m² – 7m – 30) — Actually, there's two common factors. The second one is m! Let's pull that out too!

To factor an expression, you have to break apart the middle term, so to speak. That's only possible if you can find two numbers whose product equals that of the outside terms and whose sum equals the middle term. Here, I'm just dealing with numbers and putting that variable aside.

–30 = 10 × –3

–30 = –10 × 3

–30 = –2 × 15

–30 = 2 × –15 — To solve for any potential factors, let's find all the numbers integers that multiply to –30

Now let's see which one adds up to –7!

15 – 2 = 13 — it's not this one

2 – 15 = –13 — nor this one

10 – 3 = 7 — we're pretty close! Let's switch that negative

3 – 10 = –7 — here we go! Here's our numbers!

4m[(m² – 10m) + (3m – 30)] — now we break apart the middle term. This is all multiplied by 4m, so that still encases everything with brackets.

4m[m(m – 10) + 3(m – 10)] — Factoring the two expressions

4m(m + 3)(m – 10) — simplifying to find our answer! Ta-da!

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3 years ago

A circle is translated 4 units to the right and then reflected over the x-axis. Complete the statement so that it will always be

irga5000 [103]

Answer:

The statement is now presented as:

$\exists\, (h,k)\in \mathbb{R}^{2} /f: (x-h^{2})+(y-k)^{2}=r^{2}\implies f': [x-(h+4)]^{2}+[y-(-k)]^{2} = r^{2}$

In other words, this mathematical statement can be translated as:

There is a point (h, k) in the set of real ordered pairs so that a circumference centered at (h,k) and with a radius r implies a equivalent circumference centered at (h+4,-k) and with a radius r.

Step-by-step explanation:

Let $C = (h,k)$ the coordinates of the center of the circle, which must be transformed into $C'=(h', k')$ by operations of translation and reflection. From Analytical Geometry we understand that circles are represented by the following equation:

$(x-h)^{2}+(y-k)^{2} = r^{2}$

Where $r$ is the radius of the circle, which remains unchanged in every operation.

Now we proceed to describe the series of operations:

1) Center of the circle is translated 4 units to the right (+x direction):

$C''(x,y) = C(x, y) + U(x,y)$ (Eq. 1)

Where $U(x,y)$ is the translation vector, dimensionless.

If we know that $C(x, y) = (h,k)$ and $U(x,y) = (4, 0)$ , then:

$C''(x,y) = (h,k)+(4,0)$

$C''(x,y) =(h+4,k)$

2) Reflection over the x-axis:

$C'(x,y) = O(x,y) - [C''(x,y)-O(x,y)]$ (Eq. 2)

Where $O(x,y)$ is the reflection point, dimensionless.

If we know that $O(x,y) = (h+4,0)$ and $C''(x,y) =(h+4,k)$ , the new point is:

$C'(x,y) = (h+4,0)-[(h+4,k)-(h+4,0)]$

$C'(x,y) = (h+4, 0)-(0,k)$

$C'(x,y) = (h+4, -k)$

And thus, $h' = h+4$ and $k' = -k$ . The statement is now presented as:

$\exists\, (h,k)\in \mathbb{R}^{2} /f: (x-h^{2})+(y-k)^{2}=r^{2}\implies f': [x-(h+4)]^{2}+[y-(-k)]^{2} = r^{2}$

In other words, this mathematical statement can be translated as:

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3 years ago

100 Brainliest up for grabs to person who can answer all of these with no links:

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Using a discrete probability distribution, it is found that:

a) There is a 0.3 = 30% probability that he will mow exactly 2 lawns on a randomly selected day.

b) There is a 0.8 = 80% probability that he will mow at least 1 lawn on a randomly selected day.

c) The expected value is of 1.3 lawns mowed on a randomly selected day.

<h3>What is the discrete probability distribution?</h3>

Researching the problem on the internet, it is found that the distribution for the number of lawns mowed on a randomly selected dayis given by:

P(X = 0) = 0.2.

P(X = 1) = 0.4.

P(X = 2) = 0.3.

P(X = 3) = 0.1.

Item a:

P(X = 2) = 0.3, hence, there is a 0.3 = 30% probability that he will mow exactly 2 lawns on a randomly selected day.

Item b:

$P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2 = 0.8$

There is a 0.8 = 80% probability that he will mow at least 1 lawn on a randomly selected day.

Item c:

The expected value of a discrete distribution is given by the sum of each value multiplied by it's respective probability, hence:

E(X) = 0(0.2) + 1(0.4) + 2(0.3) + 3(0.1) = 1.3.

The expected value is of 1.3 lawns mowed on a randomly selected day.

More can be learned about discrete probability distributions at brainly.com/question/24855677

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