# Angles  (Page 6/29)

 Page 6 / 29

Find an angle $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ that is coterminal with an angle measuring 870°, where $0°\le \alpha <360°.$

$\alpha =150°$

Given an angle with measure less than 0°, find a coterminal angle having a measure between 0° and 360°.

1. Add 360° to the given angle.
2. If the result is still less than 0°, add 360° again until the result is between 0° and 360°.
3. The resulting angle is coterminal with the original angle.

## Finding an angle coterminal with an angle measuring less than 0°

Show the angle with measure −45° on a circle and find a positive coterminal angle $\alpha$ such that 0° ≤ α <360°.

Since 45° is half of 90°, we can start at the positive horizontal axis and measure clockwise half of a 90° angle.

Because we can find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°:

$\text{\hspace{0.17em}}-45°+360°=315°\text{\hspace{0.17em}}$

We can then show the angle on a circle, as in [link] .

Find an angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with an angle measuring −300° such that $\text{\hspace{0.17em}}0°\le \beta <360°.\text{\hspace{0.17em}}$

$\beta =60°\text{\hspace{0.17em}}$

## Finding coterminal angles measured in radians

We can find coterminal angles    measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

Given an angle greater than $\text{\hspace{0.17em}}2\pi ,$ find a coterminal angle between 0 and $\text{\hspace{0.17em}}2\pi .$

1. Subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ from the given angle.
2. If the result is still greater than $\text{\hspace{0.17em}}2\pi ,$ subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ again until the result is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2\pi .\text{\hspace{0.17em}}$
3. The resulting angle is coterminal with the original angle.

## Finding coterminal angles using radians

Find an angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with $\text{\hspace{0.17em}}\frac{19\pi }{4},$ where $\text{\hspace{0.17em}}0\le \beta <2\pi .$

When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians:

$\begin{array}{l}\frac{19\pi }{4}-2\pi =\frac{19\pi }{4}-\frac{8\pi }{4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{11\pi }{4}\hfill \end{array}$

The angle $\text{\hspace{0.17em}}\frac{11\pi }{4}\text{\hspace{0.17em}}$ is coterminal, but not less than $\text{\hspace{0.17em}}2\pi ,$ so we subtract another rotation:

$\begin{array}{l}\frac{11\pi }{4}-2\pi =\frac{11\pi }{4}-\frac{8\pi }{4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{3\pi }{4}\hfill \end{array}$

The angle $\text{\hspace{0.17em}}\frac{3\pi }{4}\text{\hspace{0.17em}}$ is coterminal with $\text{\hspace{0.17em}}\frac{19\pi }{4},$ as shown in [link] .

Find an angle of measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that is coterminal with an angle of measure $\text{\hspace{0.17em}}-\frac{17\pi }{6}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}0\le \theta <2\pi .$

$\text{\hspace{0.17em}}\frac{7\pi }{6}\text{\hspace{0.17em}}$

## Determining the length of an arc

Recall that the radian measure     $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ of an angle was defined as the ratio of the arc length     $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ of a circular arc to the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ of the circle, $\text{\hspace{0.17em}}\theta =\frac{s}{r}.\text{\hspace{0.17em}}$ From this relationship, we can find arc length along a circle, given an angle.

## Arc length on a circle

In a circle of radius r , the length of an arc $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ subtended by an angle with measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in radians, shown in [link] , is

$\text{\hspace{0.17em}}s=r\theta \text{\hspace{0.17em}}$

Given a circle of radius $r,$ calculate the length $s$ of the arc subtended by a given angle of measure $\theta .$

1. If necessary, convert $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to radians.
2. Multiply the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ by the radian measure of $\text{\hspace{0.17em}}\theta :s=r\theta .\text{\hspace{0.17em}}$

## Finding the length of an arc

Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

1. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
2. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
1. Let’s begin by finding the circumference of Mercury’s orbit.

Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:

2. Now, we convert to radians:

what is inflation
inflation is the general price increase of goods and services in an economy.
tesfie
Inflation is the persistent rise in the general price level
T-Max
inflation is characterized by increase in the general price of goods and services. when there is too much money in circulation. increase in demand of goods pursuing fewer goods. when purchasing power of money decreases .
Ejikeme
inflation is the persistent rise general price level
Habeeb
inflation is the persistent increase in price
Machall
hi
Rafiu
yes
boston
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Ayaan
how are you
Ayaan
increase in the general level of price...
different between demand and quantity demand
No difference
MansoorAfghan
demand is the overall demand for it
MansoorAfghan
actually theres no difference
MansoorAfghan
quantity demanded is used in Equilibrium of d and s
MansoorAfghan
for evrything else u use deman
MansoorAfghan
how to calculate inflation
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price of the product increase of price substitute product as people shift to cheap one
Black
what are the methods used by trade union to increase wages of their members?
strike
Pearl
the size of the commodity
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increase demand of labour decrease supply of labour
Black
keshav
but how do they do it?
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by increasing more labour and reduced the suppliers
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they can not increase labour, they increase demand of labour.
Black
how do they increase demand for labor?
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by analyzing the market equilibrium , cost reduction and cost control , savings in time .
yash
decreasing supply of labour are achieved through training and certification that require for you to employed, you must have certificate, also trade union encouraged government to restrict migration into the country causing shortage of labour supply. Note that the aim of union is to enhance life
Black
objective of union: better working conditions, liveable wage, protect member from unfair treatment which are done through negotiations betweens representative and management. known as collective bargaining.
Black
what is the nature of economics?
economics is a social science since it seeks to solve social problem of scarcity
Jamal
main concerns is the decision individuals make on the allocation of scarce resources among the competing wants
Black
in the short run firm produce a positive as long as the price is larger than what?
what is economic
economics is the study of managing the resources in order to maximize the needs and satisfy the wants to a great extent in a regulated set-up..
One explanation for deviation when there is no impact on balance of trade
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economic s is a social science that deals with human behavior as a relationship between ends and scarce means which has alternative uses
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economic is a study of mankind in ordinary business of life
FIDELIS
economics it is the study of social science that deals with human behaviour as relationship between ends and scarce means which have alternative uses
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Good having low quality , also known as giffin goods. When income increases people shift to better quality goods . Hence having a negative effect on inferior goods rather than positive relation ( ie when income increases demand increases but not in case of inferior goods ) example wheat and bajra .
yash
What do u understand by the word ENDS in professor Lord L C Robinson definition of Economics?
I understand that ENDS is the unlimited needs of human. But we have limited resources to achieve our unlimited needs/wants. Thank you.
Midhun
variable is a factor that can change
did you understand definition of variablE
nyodb
what is monopoly
A market situation where there is a single seller of a product for the buyers.
harmony
monopoly is when you have a product in the market and only one supplier got this product so he starts rising price and dominate the market, cause this product doesn't have a competitor
Anas
A market structure characterized by a single seller, selling a unique product in the market. In a monopoly market, the seller faces no competition, as he is the sole seller of goods with no close substitute.
Jale
This is a kind of market that only one seller dominate the market without no competition of a y product or seller.
Ojo
what is production possibility curve?
Lisa
a production possibility curve is a diagram that shows two goods produced in an economy in such a way that increase in the production of one good cannot be done without the decrease on the production of the other. there are efficient points, inefficient points and unattainable points on the PPC
The
production possibility curve or frontier is useful analytical tool for illustrating the concepts of scarcity, choice and opportunity costs.
wilflay
The International Year of Soils, 2015 (IYS 2015) was declared by the Sixty-eighth session of the United Nations General Assembly on December 20th, 2013 after recognizing December 5th as World Soil Day.The purpose of the IYS is to raise awarenessworldwide of the importance of soils for food security,
what
what is indifference curve
Evet
Monopoly is the explicit right given to business or entrepreneurship by the government to operate as the only entity in the economy
wilflay
what is economic system ?
wilflay
economical matter solve this system is know as economic system
what
indifference curve is a diagram that shows the combination of only 2 commodities in such a way that each point on the indifference curve gives the same level of satisfaction
The
all consumer is equal consumtion as definite all ..that type curve is indifference curve
what
what is variable?
Rajeev
economic structure of any area is ,city or country is called economic system
Jairam
what is the difference between rational and irrational choice
Amina
help
Amina
Rational choice theory is an economic principle that assumes that individuals always make prudent and logical decisions that provide them with the highest amount of personal utility. ... Most mainstream academic assumptions and theories are based on rational choice theory.
Anas
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency.
Anas
help
Mensah
The meaning of inverse
Mensah
?
Anas
inverse what ? inverse means opposite .. like if ones going down other goes up so inverse relationship
MansoorAfghan
Definition of Inversely Related: Two variables are inversely related when an increase in one variable causes a reduction in the other variable. For example, when the price of a good increases, its quantity demanded decreases.
Anas
what is opotunity cost
salam
is an alternative forgone after the best choice have been selected.for example when you have cocacola and pepsi and you choose cocacola oportunity cost will be pepsi
Amin
explain the meaning of price cealing and price floor..?
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Opportunity cost is the cost express in terms of forgone, alternatives after a choice have been made.
cyril
A price ceiling is the maximum price a seller can legally charge a buyer for a good or service. Consumers must not pay a price above the pries ceiling
Ntui
A price floor is that is imposed to protect consumers it can not be above the equilibrium price. It's the lowest price that producers must accept from a sales
Ntui
A price ceiling is a price control mechanism issued by the government to protect the consumers from high-priced commodities. It goes to set the highest amount a seller can charge per unit quantity of his product. N.B: This price mechanism isn't binding if equilibrium exists.
harmony
A price floor is a price control mechanism that seeks to protect the sellers by setting the lowest amount a seller can receive for his products. In this case, a seller will not sell his products for any amount less than the price floor amount.
harmony
It is argued that under optimization, since there is the second-order sufficient condition, the first-order condition is not necessary. Discuss.
It is argued that in optimization the first part of second order condition appears opposite to their interpretation. Explain why you think otherwise.
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we need to understand the things that manufacturer industies need to overcome the change of price, including all the all factors
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I expect the demand curve of such a case to be less extreme almost a horizobtal line.
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Difference between extinct and extici spicies
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
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