Convert each complex number to exponential form re iθ. Use principal arguments throughout. a) 2∠(π/3) (b) 3cos(π/3)−3isin(π/3) (c) 5∠(45∘)

Answer:

θ = 30°

Angle of elevation is 30°

Step-by-step explanation:

Given;

Height of wall w = 10 ft

Length of ramp l = 20 ft

Angle of elevation = θ

Applying trigonometry;

Sinθ = opposite/hypothenuse

Where;

Height of wall w = opposite (it's opposite the angle of elevation)

Length of ramp l = hypothenuse

Sinθ = w/l

θ = arcsine (w/l)

Substituting the values;

θ = arcsine (10/20)

θ = arcsine (0.5)

θ = 30°

Angle of elevation is 30°

[G/K] divides m!, and [H/K] divides (m - 1)!.

Let G be a group with a subgroup H ≤ G, and let K = ker(λ).

We aim to show that G/K is isomorphic to a subgroup of Sym(G/H), where Sym(G/H) denotes the symmetric group on the set of cosets G/H.

To establish this, we define a map from G/K to Sym(G/H) as follows:

ϕ: G/K → Sym(G/H)

ϕ(gK) = λ(g) for all g ∈ G

In other words, for each coset gK in G/K,

we assign the left multiplication by g as a permutation on the cosets of G/H.

We need to verify that ϕ is a well-defined homomorphism, injective, and surjective.

1. Well-defined:

Suppose gK = hK, i.e., g and h represent the same coset of H.

Then g\(h^{(-1)\) ∈ K (by the definition of K).

We have:

ϕ(gK) = λ(g) and ϕ(hK) = λ(h)

Since g\(h^{(-1)\) ∈ K = ker(λ),

λ(g\(h^{(-1)\)) = e, the identity element of Sym(G/H).

Therefore, ϕ(gK) = λ(g) = λ(h) = ϕ(hK).

Thus, ϕ is well-defined.

2. Homomorphism: For any g, h ∈ G, we have:

ϕ((gK)(hK)) = ϕ((gh)K)

                 = λ(gh)

                 = λ(g)∘λ(h)

                 = ϕ(gK)∘ϕ(hK)

Therefore, ϕ is a homomorphism.

3. Injective:

Let gK, hK ∈ G/K.

If ϕ(gK) = ϕ(hK), then λ(g) = λ(h).

This implies g\(h^{(-1)\) ∈ ker(λ) = K.

Therefore, gK = hK, and ϕ is injective.

4. Surjective: For any λ(g) ∈ Sym(G/H), ϕ(gK) = λ(g).

Therefore, ϕ is surjective.

Since ϕ is a well-defined homomorphism that is injective and surjective, it is an isomorphism from G/K to a subgroup of Sym(G/H).

Now, let's consider the order of G/K. By Lagrange's theorem, the order of G/K divides the order of G, denoted as |G|.

Since |G/H| = m and each coset G/H contains |H| elements,

|G| = m |H|.

Therefore, |G/K| divides m!.

Moreover, since H is a subgroup of G and |G/H| = m, the order of H divides the order of G, i.e., |G| = n |H|.

Therefore, |H/K| divides (m - 1)!. Since |H/K| is the index of H/K in G/K, denoted as [H/K], we have [H/K] divides (m - 1)!.

Learn more about Homomorphisms here:

https://brainly.com/question/6111672

#SPJ4

The question attached here seems to be incomplete, the complete question is:

Let G be group with subgroup H <= G and let K = ker(lambda) as in the previous problem. Suppose H has finite index in G, and write m = [G / H] Show that G/K is isomorphic to a subgroup of Sym(G/H). Use this to show that [G / K] divides m !, and then use this to show that [H / K] divides (m - 1)

The relationships between different angles and sides in a triangle.

\(30+60=90\)

Complementary angles are important in geometry because they often appear in problems involving right triangles and trigonometry.

A right angle is an angle that measures exactly 90 degrees, and it is usually denoted by a small square placed at the vertex where the two sides of the angle meet.

Two angles are said to be complementary if their sum is equal to 90 degrees.

For example, if one angle measures 30 degrees, the complementary angle would measure 60 degrees

\( (30 + 60 = 90). \)
1. Angle A measures 45 degrees, and angle B measures 45 degrees.

Together, they add up to 90 degrees, so they are complementary angles.
2. Angle C measures 20 degrees, and angle D measures 70 degrees.

Although they have different measures, they add up to 90 degrees, so they are also complementary angles.

for such more questions on Complementary angles

https://brainly.com/question/15168727

#SPJ11

There are several different sampling methods that could be employed for this study, including:

Simple random sampling: This method involves randomly selecting a certain number of students from each campus and measuring the distance between their hometowns and their campuses.

Stratified random sampling: This method involves dividing the student population at each campus into different strata (e.g. by major or year in school) and then randomly selecting a certain number of students from each stratum.

Cluster sampling: This method involves randomly selecting a certain number of campuses and then measuring the distance between the hometowns of all students at those campuses and their respective campuses.

Multi-stage sampling: This method involves using a combination of the above methods, such as first using cluster sampling to select a certain number of campuses and then using stratified random sampling to select a certain number of students from each campus.

This study would be an enumerative study because it seeks to measure the characteristics of a population, in this case, the average distance between the hometowns of students and their campuses, by counting and measuring them.

It is not an analytic study because it doesn't involve testing a hypothesis or making causal inferences.

To know more about types of sampling visit :

https://brainly.com/question/12902833?referrer=searchResults

#SPJ4

Step-by-step explanation:

6x^2y^3/2x-2

=6x^2y^3/2 (x-1)

=3x^2y^3/(x-1)

Answer:

3x4y3 just got this on a test and it was right

Step-by-step explanation:

Answer:

Give the triangle area so we can solve it for you.

Step-by-step explanation:

½*base*height

Step-by-step explanation:

the triangle is not given so let us take its base=b,and height=h

by using the formula=½*base*height

½*b*h

Answer:

18 cm each side

Step-by-step explanation:

Square has 4 sides. So, 72/4=18

18 cm each side

plz mark me as brainliest.

The correct answer is option c) the variability of the observed y-values around the predicted y-values.

The estimate's standard error, denoted by \(s_e\), in a regression model, is a measure of the standard deviation of the errors. The estimate's standard error is a measure of the average error deviation, or the difference between the y-values predicted by the multiple regression model and the y-values in the sample. The standard deviation of the errors/residuals is the standard error of the estimate for the regression model.

As a result, the estimate's standard error measures the variability of the observed y-values around the predicted y-values.

To learn more about standard error refer here

https://brainly.com/question/13179711

#SPJ4

Here is the complete question.

The standard error of the estimate measures: . multiple choice.

a.  the variability of the explanatory variables

b.  the variability of the values of the sample regression coefficients

c.  the variability of the observed y-values around the predicted y-values d.  the variability of the predicted y-values around the mean of the observed y-values

Answer: Let's call the number of sandals manufactured in a typical month x, and the number of running shoes manufactured in a typical month y. The total cost for sandals and running shoes in a typical month is $2450, so the following equation represents the cost:

2.5x + 4y = 2450

During the month of April, the company doubled the number of sandals manufactured, so x becomes 2x. The total cost during the month of April is $3700, so the following equation represents the cost:

2.5(2x) + 4y = 3700

Expanding the left side:

5x + 4y = 3700

Subtracting the first equation from the second equation:

5x - 2.5x + 4y - 4y = 3700 - 2450

2.5x = 1250

Dividing both sides by 2.5:

x = 500

Substituting x = 500 into the first equation:

2.5 * 500 + 4y = 2450

Expanding and solving for y:

1250 + 4y = 2450

1250 - 1250 + 4y = 2450 - 1250

4y = 1200

Dividing both sides by 4:

y = 300

So the company makes 500 pairs of sandals and 300 pairs of running shoes during a typical month.

Step-by-step explanation:

Answer:

if you're trying to find the x and y intercepts x is (-4/5,0)

Answer:

-13.5

Step-by-step explanation:

trust me

Answer:

Interrater reliability between the two data collectors was high

Step-by-step explanation:

Explanation-

Answer: 4 + 2w

Step-by-step explanation:

The height of the tomato plant in inches after w weeks will be the value of the current height plus the weekly growth pee week. This can be represented in an equation as:

h = 4 + 2(w)

h = 4 + 2w

For example, the height of the tomato plant in inches after 5 weeks will be:

h = 4 + 2w

h = 4 + 2(5)

h = 4 + 10

h = 14 inches

The house was in the shape of an octagon, with eight separate bedrooms on each side.

The house described in the statement has an octagonal shape. An octagon is a polygon with eight sides, characterized by its eight equal angles. In this case, each side of the house corresponds to one of these angles, resulting in eight separate bedrooms on each side of the octagonal structure.

The choice of an octagon as the shape of the house can have various implications. Octagonal buildings are often admired for their unique architectural design and aesthetic appeal. The shape provides symmetry and balance, allowing for an interesting and visually pleasing structure. Additionally, an octagonal layout can offer practical advantages in terms of space utilization and room distribution. In the case of the house described, each side of the octagon accommodates a separate bedroom, providing privacy and ample living space for its occupants. The symmetrical arrangement of the bedrooms around the central area of the house could create a harmonious and well-organized living environment. Overall, the octagonal shape of the house with eight separate bedrooms on each side offers both architectural and functional benefits.

Learn more about octagon here: https://brainly.com/question/12744685

#SPJ11

Mason and her two friends paid a total of $92.97, including tax, for their skiing trip. The cost includes the tickets, handwarmers, and the two hot fudge brownie sundaes.

To calculate the total amount paid by Mason and her friends, we need to consider the cost of tickets, handwarmers, and the hot fudge brownie sundaes, along with the tax.

The tickets cost $25.25 each, so for three people, the total cost for tickets is $25.25 * 3 = $75.75.

They bought 6 handwarmers at $0.75 each, resulting in a cost of $0.75 * 6 = $4.50.

The two hot fudge brownie sundaes cost $5.45 each, totaling $5.45 * 2 = $10.90.

To calculate the tax, we add up the costs of the tickets, handwarmers, and sundaes, and then multiply by the tax rate of 9%. Tax = ($75.75 + $4.50 + $10.90) * 0.09 = $9.81.

Finally, we add the costs and tax together: $75.75 + $4.50 + $10.90 + $9.81 = $100.96.

Learn more about skiing here : brainly.com/question/30857369

#SPJ11

Answer:

D)3

Step-by-step explanation:

The function is V = 35,500 * 0.75^t and the value in 7 years is $4739

From the question, we have the following parameters that can be used in our computation:

Initial = 35500

Rate = 25%

The exponential function to represent this situation can be represented as:

V = 35,500 * (1 - 0.25)^t

V = 35,500 * 0.75^t

To find out the value of the car in 7 years, we can plug in t = 7 into the exponential function:

So, we have

V = 35,500 * 0.75^7

Evaluate

V = 4739

The value of the car in 7 years will be $4739

Read more about exponential function at

https://brainly.com/question/2456547

#SPJ1

Answer:

1710 trees up to 2016 (not including 2016)

2100 trees up to 2016 (including 2016)

Step-by-step explanation:

Given that

Farm started with 30 trees in 2007.

Every year 40 more trees than the previous year.

It is actually an Arithmetic Progression (AP) with

First term, a = 30 and

Common Difference, d = 40

The AP will look like:

30, 70, 110, 150, ....

We have to find the sum of this AP upto 9 terms and 10 terms:

9 terms sum will give us the total number of trees up to 2016 (not including 2016).

10 terms sum will give us the total number of trees up to 2016 (including 2016).

Formula for sum of 'n' terms of an AP:

\(S_n=\dfrac{n}{2}(2a+(n-1)d)\\\Rightarrow S_9=\dfrac{9}{2}(2\times 30+(9-1)40)\\\Rightarrow S_9=\dfrac{9}{2}(60+320)\\\Rightarrow S_9=\dfrac{9}{2}(380)\\\Rightarrow S_9=9 \times 180 = 1710\)

\(S_{10}=\dfrac{10}{2}(2\times 30+(10-1)40)\\\Rightarrow S_{10}=5(60+360)\\\Rightarrow S_{10}=5 \times 420\\\Rightarrow S_{10}=2100\)

1710 trees up to 2016 (not including 2016)

2100 trees up to 2016 (including 2016)

In order to expand and simplify an expression, we need to multiply out the brackets and then simplify the resulting expression by collecting the like terms. Expanding brackets is the process by which we remove brackets. It is the reverse process of factorization.

To expand and simplify a binomial, use the distributive property. The distributive property states that for any two binomials, the product of each term in the first binomial and each term in the second binomial should be added together to get the expanded form of the expression.

For example, the binomial (2x + 3)(4x + 7) can be expanded using the distributive property by multiplying each term of the first binomial with each term in the second binomial. This gives us:

(2x × 4x) + (2x × 7) + (3 × 4x) + (3 × 7)

And simplifying this expression yields 8x² + 14x + 21.

Learn more about binomial numbers at

https://brainly.com/question/30100270

#SPJ4

Answer:

D

Step-by-step explanation:

i took the test

The equivalent expressions of \((x^{6} )^{-3}\) will be \((x )^{-18}\).

Those expressions who might look different but their simplified forms are same expressions are called equivalent expressions.

To derive the equivalent expressions of some expression, we can either make it look more complex or simple. Usually, we simplify it.

The given expression is;

\((x^{6} )^{-3}\)

Required to Determine the equivalent expression.

First, we need to evaluate the expression in the bracket:

Apply the following law of indices:

\((x^a)^b = x ^{ab}\)

So:

\((x^{6} )^{-3}\)

becomes

\((x )^{6 \times -3}\\\\\)

\((x )^{-18}\)

Hence: the equivalent expressions of \((x^{6} )^{-3}\) will be \((x )^{-18}\).

Learn more about expression here;

https://brainly.com/question/14083225

#SPJ5

Given:

Principal Amount, P = $4,400

Interest rate, r = 90%

Here, Camilla needs to create an exponential equation that allows her tract the interest earned over time.

This is a Compound interest problem.

To find create an exponential equation, apply the formula below:

WHere:

P is the principal amount = $4,400

r is the rate = 90% = 0.90

t is the time (number of years)

Input values into the equation above:

Therefore, the exponential equation that allows her to track the interest earned over time is:

Plugging the value for the number of years for t, you will get the interest earned over time, t.

ANSWRER:

=============================================================

Explanation:

There are n = 7 sides to this polygon (so it's a heptagon)

The interior angles add up to 180(n-2) = 180*(7-2) = 900 degrees.

So that means,

69+284+62+62+242+81+a = 900

800+a = 900

a = 900-800

a = 100

Answer:

65

Step-by-step explanation:

Answer: p=−5

Step-by-step explanation:

Answer:

p-10/-2=0

p-5=0

p=5

solved

Answer:

2(m + 5)

Step-by-step explanation:

Twice means multiplication.

2 *

Sum means addition.

m + 5

2(m + 5)

Best of Luck!

There are 56 different ways to select three different integers from 1 to 10 such that no two chosen integers are consecutive.

To determine the number of ways to select three different integers from 1 to 10 such that no two chosen integers are consecutive, we can use combinatorics principles.

First, let's consider the total number of ways to choose three integers without any restrictions. This can be calculated using the combination formula:

C(n, r) = n! / (r!(n-r)!)

In this case, we have 10 integers to choose from, and we want to choose 3. Therefore, the total number of ways to choose three integers without any restrictions is:

C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = 120

Now, let's consider the restricted condition that no two chosen integers can be consecutive. We can approach this by selecting three non-consecutive integers from the available set.

To do this, we can start by choosing any integer, then excluding its two adjacent integers. For example, if we choose 1, we cannot choose 2 or 3. Similarly, if we choose 10, we cannot choose 9 or 8.

Since there are two adjacent integers to exclude for each choice, we have 8 integers remaining to choose from. Therefore, the number of ways to select three non-consecutive integers is:

C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = 56

To learn more about combination click on,

https://brainly.com/question/24344074

#SPJ4

Answer:

1.-4,-1,1,6

2.by substituting the values given, we get

32-|15+8|

32-|23|

32-23=9

3.-15+7=8

4..27-(-12)=27+12=39

5.-6.05+(-2.1)

-6.05-2.1=-8.15

6.-3/4-(-2/5)

-3/4+2/5

=-7/20

7.-9(-12)

=108

8.(3.8)(-4.1)

-15.58

9.(-8x)(-2y)+(-3y)(z)

(16xy)+(-3zy)

(16xy)-(3zy)

10.by substituting the value given,we get

2.5*(-3.2)+5

-8+5

-3

so here are the answers,mark as brainliest if u find it useful

thank you

Answer:

by my calculator I got 2.25