Let `z= cos theta +i sin theta` .Then the value of `sum_(m=1)^15` `lm(Z^(2m-1))` at `theta=2` deg is

You need to evaluate the sum of imaginaries of the given powers of complex number z, such that:

`Sigma_(m=1)^15 Im(z^(2m-1)) = Im(z^(2*1-1)) + Im(z^(2*2-1)) + Im(z^(2*3-1)) + ... + Im(z^(2*15-1))`

`Sigma_(m=1)^15 Im(z^(2m-1)) = Im(z^1) + Im(z^3) + Im(z^5) + ... + Im(z^29)`

All the powers of the complex number z, may be evaluated with De Moivre formula, such that:

`z^n = (cos theta + i*sin theta)^n => z^n = (cos (n*theta) + i*sin (n*theta))`

`z^3 = cos (3theta) + i*sin(3 theta)`

........

`z^29 = cos (29theta) + i*sin(29 theta)`

The imaginary part of the sum is `Im(z^1+z^3+...z^29) = cos theta + cos (3theta) + ... + cos(29 theta)`

You may group the terms such that:

`cos theta + cos (29 theta) = 2cos ((theta + 29theta)/2)*cos((theta - 29theta)/2)`

`cos theta + cos (29 theta) = 2cos (15theta)cos(-14theta)`

Since `cos(-theta)=cos theta`

`cos theta + cos (29 theta) = 2cos (15theta)cos(14theta)`

You may calculate the next sum `cos (3 theta) + cos (27theta),` such that:

`cos (3 theta) + cos (27theta) = 2cos (15theta)cos(12theta)`

`cos (5 theta) + cos (25 theta) = 2cos (15theta)cos(10theta)`

`cos (7 theta) + cos (23 theta) = 2cos (15theta)cos(8theta)`

`cos (9 theta) + cos (21 theta) = 2cos (15theta)cos(6theta)`

`cos (11 theta) + cos (19 theta) = 2cos (15theta)cos(4theta)`

`cos (13 theta) + cos (17 theta) = 2cos (15theta)cos(2theta)`

Notice that the term `cos (15 theta)` remains. You may take out the common factor `cos (15theta)` such that:

`Im(z^1+z^3+...z^29) = cos (15 theta)(2cos(12theta) + 2cos(10theta) + 2cos(8theta) + 2cos(6theta) + 2cos(4theta) + 2cos(2theta) + 1)`

You may put now` theta = 2^o` , such that:

`cos (15*2^o) = cos 30^o = sqrt3/2`

You may group again the terms, such that:

`2cos(12theta) + 2cos(2theta) = 4 cos (7 theta)*cos (5 theta)`

`2cos(10theta) + 2cos(4theta) = 4 cos (7 theta)*cos (3 theta)`

`2cos(8theta) + 2cos(6theta) = 4 cos (7 theta)*cos ( theta)`

Factoring out 4` cos (7 theta)` yields:

`2cos(12theta) + 2cos(10theta) + 2cos(8theta) + 2cos(6theta) + 2cos(4theta) + 2cos(2theta) = 4 cos (7 theta)*(cos (5 theta) + cos (3 theta) + cos ( theta))`

You may group again the terms, such that:

`cos (5 theta) + cos ( theta) = 2cos (3theta)*cos (2theta)`

`2cos (3theta)*cos (2theta) + cos (3theta) = cos(3theta)(2cos(2theta) + 1))`

Using the formula of double angle yields:

`cos (2theta) = 2cos^2 theta - 1`

`cos(3theta)(2cos(2theta) + 1)) = cos(3theta)(4cos^2 theta -2 + 1))`

`cos(3theta)(2cos(2theta) + 1)) = cos(3theta)(4cos^2 theta -1))`

`2cos(12theta) + 2cos(10theta) + 2cos(8theta) + 2cos(6theta) + 2cos(4theta) + 2cos(2theta) = 4 cos (7 theta)*cos(3 theta)*(4cos^2 theta -1))`

`Im(z^1+z^3+...z^29) = cos (15 theta)(4 cos (7 theta)*cos(3 theta)*(4cos^2 theta -1)) + 1)`

Hence, evaluating `Sigma_(m=1)^15 Im(z^(2m-1)) = (sqrt3/2)*)(4 cos (14^o)*cos(6^o)*(4cos^2(2^o) -1)) + 1).`

In Fahrenheit 451 by Ray Bradbury, Montag enters his home after his first meeting with Clarisse. He stares at the blank wall, but in his memory,...

After meeting Clarisse in the beginning of Fahrenheit 451, Montag goes home and looks at a blank wall. However, the memory of Clarisse is so fresh in his memory that he can picture her perfectly. The following extended simile illustrates this:

She had a very thin face like the dial of a small clock seen faintly in a dark room in the middle of the night when you waken to see the time and see the clock telling you the hour and the minute and the second, with a white silence and a glowing, all certainty and knowing what it has to tell of the night passing swiftly on toward further darknesses, but moving also toward a new sun. (8)

This description is profound because it is thorough and accessible; most people can imagine what he's describing and picture it clearly in their mind's eye. The fact that he compares her face to a clock at night and muses about the clarity of moving towards a new sun (or starting a new day) suggests that Clarisse excites a part of Montag that has been dead or unexplored for a long time.

Montag further compares her face to a mirror because he feels that she understands him, and this is a rare feeling: "How rarely did other people's faces take of you and throw back to you your own expression, your own innermost trembling thought?" (8). He senses that she has an "incredible power of identification," and wonders if his eyelid itched if hers would blink. Their meeting seems meant to be for Montag.

Why do effervescent tablets dissolve in water? Why do they dissolve faster in hot water? Why do effervescent tablets dissolve slower in carbonated...

The chemical reaction that occurs when effervescent tablets are placed in water is as follows:

Reaction in Words: 

Citric Acid + Sodium Bicarbonate yields Water + Carbon Dioxide + Sodium Citrate

Reaction in Symbols:

~C_6H_8O_7 + ~3NaHCO_3  -> ~3H_2O + ~3CO_2 + ~Na_3C_6H_5O_7

How the Reaction Occurs:

• Before the tablet is dissolved, the citric acid and sodium bicarbonate are in solid form and unable to react with one another.

• In order for the citric acid and sodium bicarbonate to react, they must be dissolved in water. Dissolving breaks the two chemicals into their positive and negative ions. This enables them to come in contact and react with one another.

Why do effervescent tablets dissolve in water?

• Citric acid is an acid and sodium bicarbonate is an ionic substance. Both of these types of substances are soluble in water. When acids and ionic substances are placed in water, the substances break apart into positive and negative ions. Water is a polar molecule. This means it has a positive side and a negative side. These substances are able to dissolve when the positive side of the water molecules interacts with the negative ions and the negative side of the water molecule interacts with the positive ions. When a substance has been dissolved in water, it is called an aqueous solution. 

Why do effervescent tablets dissolve faster in hot water?

• In order for a reaction between two substances to occur, the substances must come in contact with one another. Anything that increases the probablity that the substances will come in contact with one another will increase the rate of the reaction. As temperature increases, the movement of the substance's particles increases. This makes it more likely that they will come in contact with one another and react. The more the substances are able to come in contact, the faster the reaction will occur.

Why do effervescent tablets dissolve slower in carbonated beverages?

• The ingredients in effervescent tablets are less soluble in carbonated beverages than in water. Water is often called "the universal solvent" because it is so good at dissolving ions due to its positive and negative sides. Since effervescent tablets are less soluble in carbonated beverages, it takes longer for the tablets to dissolve and enable the reaction to occur.

`int (x + 1) sqrt(2x + x^2) dx` Evaluate the indefinite integral.

You need to use the following substitution  `2x + x^2= t` , such that:

`2x + x^2= t=>(2 + 2x)dx = dt => (x + 1) dx = (dt)/2`

`int (x+1)sqrt(2x+x^2)dx= (1/2)int sqrt t dt`

`(1/2)int sqrt t dt = (1/3)t^(3/2) + c`

Replacing back `2x + x^2` for t yields:

`int (x+1)sqrt(2x+x^2)dx = (1/3)(2x + x^2)^(3/2) + c`

Hence, evaluating the indefinite integral, yields `int (x+1)sqrt(2x+x^2)dx = (1/3)(2x + x^2)^(3/2) + c`

How is the shadow of an object affected by its distance from a light source?

In order to understand how an objects distance from the light source effects its shadow, lets first talk about what causes a shadow. A shadow is caused when an object lies in-between a light source and the background on which the light is landing. The object blocks a portion of light from the light source from making it to the final point on the background object, leading to a shadow.

To understand how the shadow is effected by the objects distance from the light source one must use principals of geometry. Thinking about it in this way, one can realize that the closer an object is to the source of light, the larger the shadow it casts. This is because the closer an object is to the light source, the greater area of the light the object will block, increasing shadow size. Alternatively, the further an object is located from the light source the less area it will block, leading to a smaller shadow being cast.

Another factor in shadow length is angle of the light source. The closer to horizontal the angle is relative to the object casting the shadow, the longer the shadow will be. Oppositely, the closer the light source is to completely vertical, relative to the object, the smaller the shadow will be cast. Hope this helps! 

In "A Christmas Memory" by Truman Capote, why does Buddy's friend remain unnamed?

In Truman Capote’s short story “A Christmas Memory” Buddy’s cousin/friend remains unnamed. This allows for Buddy to keep his memory of her personal and close to his heart. Her name is not necessary, her relationship with Buddy is. He gives a thorough description of her age, looks, and personality traits. She is in her sixties, does not care much about her appearance, is deeply religious, brave, and child-like. She does not need a name; her image makes her more relatable yet allows Buddy to keep her for himself. She is an enigma and a symbol of his childhood. It also provides a bit of mystery to the reader as to whom the woman is. The other adults in the story also remain unnamed but they are described as being uncaring and obtrusive.

Truman Capote did reveal that he based the cousin’s character on Miss Sook Faulk, one of his mother’s distant relatives with whom he spent some of his early years. Perhaps when the story was written, Capote tried to save her identity only revealing it later. Miss Faulk shared attributes with the character in the story. His mother left him in the care of relatives during some tumultuous years for the family.

An increase in the money wage rate decreases aggregate supply and shifts the aggregate supply curve leftward. A fall in the money wage rate lowers...

The answer to this depends on how, exactly, you are using the term “price level.”  If you are talking about prices in the economy as a whole, the price level causes a movement along the aggregate supply (AS) curve.  If you are talking about the price of labor and other resources that go into producing goods and services, the price level causes the AS curve to shift.

The AS curve shows what level of output an economy can produce at any given price level.  This means that the horizontal axis is measured in units of real Gross Domestic Product while the vertical axis shows the price level (GDP deflator) for the economy as a whole.  Given this information, we can see that when the prices in the economy as a whole change (when the GDP deflator changes), we are just moving along a given AS curve.

However, the price level of resources is something that will cause the AS curve to shift.  It is one of the major determinants of aggregate supply.  When the price of resources like labor or oil changes, the amount that producers are willing and able to produce changes as well.  This stands to reason because a manufacturer who has to pay more for labor and other resources will not make as much profit and will therefore not want to produce as many goods or services.  Please note, however, that this only applies to the short-run AS curve.  In the long run, resource prices do not affect the AS curve.

The most likely answer here is that the price level causes a movement along the AS curve because “price level” usually refers to the GDP deflator or other measures of general prices in an economy.  However, if you are using “price level” to refer to the price of resources, it can cause a shift in the short run AS curve.