Self harm in adolescents Essay Example | Topics and Well Written Essays - 2000 words

Self damage in teenagers - Essay Example It is important to comprehend the moral contemplations when managing a pre-adult who means or is taking part in self injury. There are moral issues concerning the understudy. The understudy ought to be guaranteed of classification on the issues they talk about with the advisor or medical attendant. Moreover, there are moral worries on parent’s obligation and the schools duty. The guide ought to keep up exclusive requirements of polished skill. It is significant that the advisor is fit for recognizing the manifestations of self-injury early. Dominant part of young people who take part in the self damaging acts do so when they are green beans or while encountering incredible enthusiastic difficulties. They guarantee that they can't be found and may proceed for long on the off chance that they don't get helped (Motz, 2009). The school specialists and guardians should benefit the assets important to support security. As people help the self damaging youthful, it is important to know about the moral difficulties. Data about the self damaging pre-adult ought to stay classified, except if the data would keep risk from the pre-adult from making further mischief themselves or others. Self harmful conduct ought to be accounted for in any event, when the juvenile isn't presented to risk, yet needs help from experts or guardians. Notwithstanding, self damaging conduct ought not be considered as a marker of self-destructive goal. For certain young people, harming self diminishes their pressure and encourages them manage pressure. Talking with lawful specialists just as the school organization on the issue is important. A broad and serious evaluation of circumstance is fundamental. This incorporates; setting up the nature and recurrence of oneself caused hurt. There is have to counsel if the juvenile is harming an d needs care. On the off chance that they are not coordinating and increment the recurrence of harming self, counseling the expert could be important to empower them adapt to their difficulties. The legitimate specialists might be associated with building up the idea of injury, in the event that it is brought about by the youthful or by someone else. Clinical mediation might be vital. The juvenile could be having wounds or disease that needs treatment. Comprehend the feelings that cause the youthful to hurt self and conceivable inspiration for their activities. Enquire on the off chance that they dispense their injuries or mischief when alone or when with others. Make certain to know whether they share objects of injury with others. Research what triggers the compelling feelings. In addition, discover who thinks about the

Business Law Case Essay Example | Topics and Well Written Essays - 750 words

Business Law Case - Essay Example odgers for causing her, deliberately, enthusiastic pain and anguish however the litigants guarantee that no harm is recoverable except if trouble bring about physical injury. Punishment of passionate pain exists if a â€Å"extreme and incredible direct deliberately or recklessly† brings about extreme enthusiastic insecurity in another gathering (Mann and Rogers 120). An individual that causes such a trouble accept obligation for the pain and potential results of the misery. A demonstration is additionally supposed to be crazy in the event that it dismisses potential results or comes up short. As per the third repetition, information on conceivable extreme enthusiastic shakiness and inability to take measures for alleviating impacts of the precariousness and lack of interest over the potential results characterize carelessness. An individual is qualified for harms for serious enthusiastic misery is endured passionate flimsiness is extreme and if a conventional individual would endure the results under ordinary conditions. There is no requirement for verification of physical mischief so as to recuperate harms for enthusiastic pain (Mann and Rogers 12 0). The instance of Ferrell v. Mikula 627 SE2d7 shows the standard. For the situation, an administrator at Ruby Tuesday eatery requested a security offers to follow clients who were accepted suspected to have left without paying for their requests. This was anyway a misstep in light of the fact that the objective had taken care of their tabs and were mistaken for two gatherings who had been pardoned from making installments. The objective were then halted, bound and set in police watch vehicle however later discharged after affirmation that they had really taken care of their tabs. The objective sued for punishment of passionate misery that was excused at the first and investigative court. The court of intrigue of Georgia contended that a case for punishment of enthusiastic misery must meet four components. There more likely than not been deliberate of wild conduct that must further be â€Å"extreme or outrageous† (Mann and Rogers 121). The demonstration must have further

Children Growing Up Essay Example

Children Growing Up Essay Example Children Growing Up â€" Essay Example > The paper “ Children Growing Up” is a worthy example of an essay on family and consumer science. It is quite evident that modern lifestyles have greatly changed the face of family life with the main focus on children up growing. Children are growing up at a very fast rate and therefore there is a need for the larger society to protect children for the sake of the world’ s future generation. Today, modern lifestyles imply that most parents have little time for children. Due to this reason, many children suffer because they do not get as much attention from their parents like the way children in the past did. Due to this, researches indicate that children are struggling to make friends at school so as to fill the gap created by their parents (Fletcher et al 2008). Further, modern lifestyles where parent spends a lot of time working has damaged children social skills thus fueling the feeling of loneliness among the children generation. Modern lifestyles have created great concerns where children are exposed to a toxic mix of modern life where they are exposed to junk food as well as electronic entertainment. Rather than being with their children, parents have increased the use of MP3 players, internet and mobile phone making children unable to interact greatly with their peers. It is quite evident that the inability of children to interact with their peers can result in a struggle for these children when forming a relationship in later life (Fletcher et al. 2008). Fletcher et al (2008) maintain that modern lifestyles have greatly increased the prevalence of obese as well as overweight children especially those that lack parental attention. Concluding remarks from experts indicate that modern lifestyles adopted by parents are and will continue to have a damaging effect on children leading to depression, developmental conditions and behavioral problems such as attention deficit hyperactivity and autism. It is therefore correct to state that modern lifestyles mean that many people have little time for children, thus many children suffer because they do not get as much attention from their parents as children did in the past. ReferencesFletcher, A., et al.2008, Parenting style as a moderator of associations between maternal disciplinary strategies and child well-being. Journal of Family Issues, 29(12), 1724-1744.

Admiral Lord Thomas Cochrane in the Napoleonic Wars

The Expedition Of Lewis And Clark - 1270 Words

Without the expedition of Lewis and Clark, the American history that people know would be completely different changing the accomplishments in history. The background and experience Lewis and Clark already developed prior to the expedition would be very beneficial and help make the journey easier. The development of the mission played a key part in the expedition itself because if the development did not play out exactly how Jefferson planned, then the expedition may not have been as successful. A big part of history, the expedition was very significant and impacted American society in ways that no one will ever be able to understand. Although the expedition of Lewis and Clark would be a very long journey, the United States would not have developed into such a prosperous country without the two men who traveled across the terrain in rain and snow in order to improve the United States. Meriwether Lewis was born August 18, 1724 near Ivy Virginia where he would begin a journey that woul d change history. In 1801, Thomas Jefferson asked Lewis to be his private secretary after he helped in the Whiskey Rebellion, which was farmers who started an uprising against taxes in 1794. Lewis had a very prominent role within the state militia, which would benefit him in the future. William Clark was born August 1, 1770 in Caroline County Virginia where he would influence America in many ways. At the young age of 19, Clark entered into the military where people knew him as the youngerShow MoreRelatedThe Lewis And Clark Expedition1333 Words   |  6 PagesMeriwether Lewis. Lewis turned for assistance and invited the U.S soldier and experienced outdoorsman, William Clark, to share command of this legendary expedition. The explorers marched to the unexpected challenges and experiences that laid within the unfamiliar land. Their contributions to the extraordinary expedition provided valuable information regarding the topography and ecology the Louisiana territory had to offer. Appointed for the Position The Lewis and Clark Expedition coveredRead MoreLewis and Clark Expedition1018 Words   |  4 PagesJames Jang William and Meriwether, better known as Lewis and Clark, were hired by the U.S. president Thomas Jefferson to explore the newly bought Louisiana Purchase. The Louisiana Purchase was bought from France in 1803. Lewis and Clark started their expedition in 1804 near St. Louis. The group of explorers called themselves the corps of discovery. In the first winter they were helped by a Shoshone Indian named Sacagawea. Their journey was full of trouble and challenges but in the end led to a greatRead MoreThe Lewis And Clark Expedition1405 Words   |  6 Pages Sacajawea is a renowned Native American woman who played a significant role in the Lewis and Clark Expedition. As stated by Brown, â€Å"She turned to dreaming of the future, of what it might hold for Pomp (her son), as she had dreamed of what might lie ahead for her when she had learned she was to be going on the expedition. But it was no dream that she had become a part of history† (110). Although most of her history is unc lear, one thing is definite - without the help of Sacajawea, the United StatesRead MoreThe Lewis And Clark Expedition1415 Words   |  6 PagesCaptain Meriwether Lewis and William Clark put their lives on the line only to return with information regarding the Pacific Northwest of the United States territory. Their reasons behind this journey, being to expand the United States to the West, involved the entire nation but debatably, their accomplishments and the reaction of others became a major part of today’s history. The foremost reason behind the Lewis and Clark expedition was all based on Thomas Jefferson’s, president at the time, interestRead MoreThe Lewis And Clark Expedition1433 Words   |  6 PagesThe Lewis and Clark expedition was a truth that was to become the crowning accomplishment in the lifetime of the brilliant thinker, inventor, and founding father, Thomas Jefferson . It has become a profounding turning point throughout America’s history. Investigating the recently obtained Louisiana Territory, which nearly doubled the size of the country, arranged Jefferson the opportunity to widen the boundaries of the United States to include both the Atlantic and Pacific oceans. The threat of theRead MoreThe Lewis And Clark Expedition1424 Words   |  6 Pages On May 14, 1804 in the eastern city of Saint Louis, William Clark and Merry-weather Lewis set of on the westward adventure that would change America as we know it today. Their journey began on the Mississippi River, those rapids would propel Lewis and Clark into the Corps of Discovery. Across the vast land that these men would soon travel lived the many native-american tribes. The Native people hunted freely across their western lands, lived their life as one with the ground they so carefullyRead MoreThe Expedition of Lewis and Clark581 Words   |  2 PagesThis report is on Lewis and Clarks adventure through the United States. On May 14,1803 William Clark and nearly four dozen other men met up with Meriwether Lewis on May 20. The Lewis and Clark Expeditio,n The Corps of Discovery began making its way up the †¢. \\?\ \? Missouri on a †¢ss-toot-long ship and two smaller boats. As they traveled,Clark spent most of his time on the ship journaling the course and making maps. Lewis wa s often on the shore, studying the rock formations, soil,animalsRead MoreThe Expedition Of Lewis And Clark1522 Words   |  7 PagesMany people in America know of the historical expedition of Lewis and Clark, but only a few know of the expedition to California led by Jedediah Smith. This expedition took place after Thomas Jefferson signed the papers to acquire a huge region in the west through the Louisiana Purchase. After this expansion many people were eager to explore the west and among those was Jedediah Smith. From his childhood, Smith dreamed one day of exploring the unknown west part of the United States. He once saidRead MoreLewis And Clark Expedition : The Great Expedition2140 Words   |  9 Pages Lewis and Clark Expedition The Lewis and Clark expedition was amongst one of the first major American expeditions. This expedition helped the United States advance in science, land as well as gaining many valuable resources. The Lewis and Clark expedition is also an amazing American story. Lewis and Clark went through extraordinary situations to expedite Americas growth and science. They accomplished this expedition with strenuous encounters with the NativeRead MoreLewis and Clark Expedition Essay926 Words   |  4 PagesThis paper will explain why Lewis and Clark are two of the greatest explorers in American history. Some of the distinguishing factors of these explorers and the three main points in this paper are their exploration of the uncharted west by way of the Missouri river, the many discoveries made along the way, and the effect they had on the westward expansion of the United States. In January 1803, Thomas Jefferson sent a confidential message to Congress asking for approval and funding of the exploration

Marxist View on the Family Free Essays

Marxists Views on the Family There are three Marxists views of the family, Karl Marx, Friedrich Engels and Eli Zaretsky; they all see all institutions such as education, the media and the family as maintaining class inequality and capitalism. Marxism is a as conflict perspective as it describes a form of inequality where groups could potentially competes for power. Modern Marxist agrees that: * Families socialise children to be obedient and hardworking, which benefits capitalists; * Wealth is passed down families, perpetuating inequalities; * Families are too privatised, discouraging wage-labourers from uniting against capitalism. We will write a custom essay sample on Marxist View on the Family or any similar topic only for you Order Now Community living is preferable; * A communist society in which all means of production, such as farms and factories, are collectively owned and workers receive a fair share of the profits should replace private ownership of businesses. Marxist say the family has three main functions for capitalism: 1. Inheritance of property- Marx called the earliest classless society ‘primitive communism’ at this stage there was no such thing as family. As society developed private property became important. Engels said the patriarchal monogamous nuclear family emerged (male dominated). In order to ensure the legitimate heir inherited from them. Marx said with the overthrow of capitalism the means of production would be owned collectively so there would be no need for the nuclear family to exist as a means of transmitting private property down the generations. 2. Ideological Function (The idea that family brainwashes us into capitalism)- Marxists say the family persuades people to think of capitalism as a fair, natural and unchangeable system. Families socialise children into thinking that hierarchy and inequality are inevitable. The family prepares people to take orders at work. Zaretsky says people are encourages to think of the family as a haven from the outside world but this is largely an illusion. As even in the privacy of our home we can subject to state control. 3. A unit of consumption- The family is an important consumer of products and has a major role in generating profits for capitalism. Advertisers urge us to ‘keep up with the Jones’. They encourage ‘pester power’ from children. Children who don’t have the latest products may be bullied. Thus Marxists see the family as performing several functions for capitalist society: the inheritance of private property, socialisation into accepting inequality, and a source of profit. According to Marxists these may benefit capitalism but not members of the family. How to cite Marxist View on the Family, Essay examples

Trigonometry Essay Example

Trigonometry Essay As you see, the word itself refers to three angles a reference to triangles. Trigonometry is primarily a branch of mathematics that deals with triangles, mostly right triangles. In particular the ratios and relationships between the triangles sides and angles. It has two main ways of being used: 1. In geometryIn its geometry application, it is mainly used to solve triangles, usually right triangles. That is, given some angles and side lengths, we can find some or all the others. For example, in the figure below, knowing the height of the tree and the angle made when we look up at its top, we can calculate how far away it is (CB). (Using our full toolbox, we can actually calculate all three sides and all three angles of the right triangle ABC). 2. AnalyticallyIn a more advanced use, the trigonometric ratios such as as Sine and Tangent, are used as functions in equations and are manipulated using algebra. In this way, it has many engineering applications such as electronic circuits and mechanical engineering. In this analytical application, it deals with angles drawn on a coordinate plane, and can be used to analyze things like motion and waves. Chapter-1Angles in the Quadrants( Some basic Concepts)In trigonometry, an angle is drawn in what is called the standard position. The vertex of the angle is on the origin, and one side of the angle is fixed and drawn along the positive x-axis. Names of the partsThe side that is fixed along the positive x axis (BC) is called the initial side. To make the angle, imagine of a copy of this side being rotated about the origin to create the second side, called the terminal side. We will write a custom essay sample on Trigonometry specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Trigonometry specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Trigonometry specifically for you FOR ONLY $16.38 $13.9/page Hire Writer The amount we rotate it is called the measure of the angle and is measured in degrees or radians. This measure can be written in a short form: mABC = 54 ° which is spoken as the measure of angle ABC is 54 degrees. If it is not ambiguous, we may use just a single letter to denote an angle. In the figure above, we could refer to the angle as ABC or just angle B. In trigonometry, you will often see Greek letters used to name angles. For example the letter ? (theta), but on this site we always use ordinary letters like A,B,C. The measure can be positive or negativeBy convention, angles that go counterclockwise from the initial side are positive and those that go clockwise are negative. In the figure above, click on reset. The angle shown goes counterclockwise and so is positive. Drag A down across the x-axis and see that angles going clockwise from the initial side are negative. See Trig functions of large and negative angles The measure can exceed 360 °In the figure above click reset and drag the point A around counterclockwise. Once you have made a full circle (360 °) keep going and you will see that the angle is greater than 360 °. In fact you can go around as many times as you like. The same thing happens when you go clockwise. The negative angle just keeps on increasing. Coterminal anglesIf you have one angle of say 30 °, another of 390 °, the two terminal sides will be in the same place (390 = 360+30). These two angles would then be called coterminal angles. They would be in the same place on the plane but have different measures (30 ° and 390 °). Degrees and radiansThe measure of an angles can be expressed in degrees or radians, but in trigonometry radians are the most common. See Radians and Degrees. Recall than there are 2? adians in a full circle of 360 °, so 1 radian is approximately 57 °. In the figure above, click on radians to change units. | Standard position of an angle In trigonometry an angle is usually drawn in what is called the standard position as shown below. In this position, the vertex of the angle (B) is on the origin of the x and y axis. One side of the angle is always fixed along the positive x-axis that is, going to the right along the axis in the 3 oclock direction (line BC). This is called the initial side of the angle. The other side of the angle is called the terminal side. Initial side of an angle In trigonometry an angle is usually drawn in what is called the standard position as shown below. In this position, the vertex of the angle (B) is on the origin of the x and y axis. One side of the angle is always fixed along the positive x-axis that is, going to the right along the axis in the 3 oclock direction (line BC). This is called the initial side of the angle. The other side of the angle is called the terminal side. Terminal side of an angle In trigonometry an angle is usually drawn in what is called the standard position as shown on the right. In this position, the vertex (B) of the angle is on the origin, with a fixed side lying at 3 oclock along the positive x axis. The other side, called the terminal side is the one that can be anywhere and defines the angle. In the figure below, drag point A and see how the position of the terminal side BA defines the angle. Quadrantal Angle Definition: Angles in the standard position where the terminal side lies on the x or y axis. For example: 90 °, 180 ° etc. A quadrantal angle is one that is in the standard position and has a measure that is a multiple of 90 ° (or ? /2 radians). A quadrantal angle will have its terminal lying along an x or y axis. | Coterminal angles From co -together, terminal -end position Definition: Two angles are coterminal if they are drawn in the standard position and both have their terminal sides in the same location| Recall that when an angle is drawn in the standard position as above, only the terminal sides (BA, BD) varies, since the initial side (BC) remains fixed along the positive x-axis. If two angles are drawn, they are coterminal if both their terminal sides are in the same place that is, they lie on top of each other. In the figure above, drag A or D until this happens. If the angles are the same, say both 60 °, they are obviously coterminal. But the angles can have different measures and still be coterminal. In the figure above, rotate A around counterclockwise past 360 ° until it lies on top of DB. One angle (DBC) has a measure of 72 °, and the other (ABC) has a measure of 432 °, but they are coterminal because their terminal sides are in the same position. If you drag AB around twice you find another coterminal angle and so on. There are an infinite number of times you can do this on either angle. Either or both angles can be negative In the figure above, drag D around the origin counterclockwise so the angle is greater than 360 °. Now drag point A around in the opposite direction creating a negative angle. Keep going until angle DBC is coterminal with ABC. You can see that a negative angle can be coterminal with a positive one. How to tell if two angles are coterminal. You can sketch the angles and often tell just form looking at them if they are coterminal. Otherwise, for each angle do the following: * If the angle is positive, keep subtracting 360 from it until the result is between 0 and +360. In radians, 360 ° = 2? radians) * If the angle is negative, keep adding 360 until the result is between 0 and +360. If the result is the same for both angles, they are coterminal. Why is this important? In trigonometry we use the functions of angles like sin, cos and tan. It turns out that angles that are coterminal have the same value for these functions. For example, 30 °, 39 0 ° and -330 ° are coterminal, and so sin30 °, sin390 ° and sin(-330 °) and all have the same value (0. 5). Reference Angle: The smallest angle that the terminal side of a given angle makes with the x-axis is called reference angle. Chapter-2 Measurement Of Angles TRIGONOMETRY, as it is actually used in calculus and science, is not about solving triangles. It becomes the mathematical description of things that rotate or vibrate, such as light, sound, the paths of planets about the sun, or satellites about the earth. It is necessary therefore to have angles of any size, and to extend to them the meanings of the trigonometric functions. An angle is the opening that two straight lines form when they meet. When the straight line FA meets the straight line EA, they form the angle we name as angle FAE. Letter A, which we place in the middle, labels the point where the two lines meet, and is called the vertex of the angle. When there is no confusion as to which point is the vertex, we may speak of the angle at the point A, or simply angle A. The two straight lines that form an angle are called its sides. And the size of the angle does not depend on the lengths of its sides. We can see that in the figure above. For if the point C is in the same straight line as FA, and B is in the same straight line as EA, then angles CAB and FAE are the same angle. Now, to measure an angle, we place the vertex at the center of a circle we call that a central angle), and we measure the length of the arc that portion of the circumference that the sides intercept. We then determine what relationship that arc has to the entire circumference, which is an agreed-upon number. (In degree measure that number is 360; in radian measure it is 2?. ) The measure of angle A, then, will be length of the arc BC r elative to the circumference BCD or the length of arc EF relative to the circumference EFG. For in any circles, equal central angles determine a unique ratio of arc to circumference. (See the theorem of Topic 14. It is stated there in terms of the ratio of arc to radius, but the circumference is proportional to the radius:   C = 2? r. ) There are two systems for measuring angles. One is the well-known system of degree measure. . Degree measure To measure an angle in degrees, we imagine the circumference of a circle divided into 360 equal parts, and we call each of those equal parts a degree. Its symbol is a small 0:   1 ° 1 degree.   The full circle, then, will be 360 °. But why the number 360? What is so special about it? Why not 100 ° or 1000 °? The answer is two-fold. First, 360 has many divisors, and therefore it will have many whole number parts. It has an exact half and an exact third which a power of 10 does not have. 360 has a fourth part, a fifth, a sixth, and so on. Those are natural divisions of the circle, and it is very convenient for their measures to be whole numbers. (Even the ancients didnt like fractions) Secondly, 360 is close to the number of days in the astronomical year: 365. The measure of an angle, then, will be as many degrees as its sides include. To say that angle BAC is 30 ° means that its sides enclose 30   Ã‚  of those equal divisions. Arc BC is  |   30 60|   of the entire circumference. | So, when 360 ° is the measure of a full circle, then 180 ° will be half a circle. 90 ° one right angle will be a quarter of a circle; and 270 ° will be three quarters of a circle:   three right angles. Let us now see how we deal with angles in the x-y plane. Standard position We say that an angle is in standard p osition when its vertex A is at the origin of the coordinate system, and its Initial side AB lies along the positive x-axis. We say that AB has swept out the angle BAC, and that AC is its Terminal side. We now think of the terminal side AC as rotating about the fixed point A. When it rotates in a counter-clockwise direction, we say that the angle is positive. But when it rotates in a clockwise direction, as AC, the angle is negative. When the terminal side AC has rotated 360 °, it has completed one full revolution. Problem 1. How many degrees corresponds to each of the following? To see the answer, pass your mouse over the colored area. To cover the answer again, click Refresh (Reload). a)   A third of a revolution     Ã‚   A third of 360 ° = 360 ° ? 3 = 120 ° b)   A sixth of a revolution     Ã‚   360 ° ? 6 = 60 ° c)   Five sixths of a revolution     Ã‚   5 ? 60 ° = 300 ° d)   Two revolutions     Ã‚   2 ? 60 ° = 720 ° e)   Three revolutions     Ã‚   3 ? 360 ° = 1080 ° f)   One and a half revolutions     Ã‚   360 ° + 180 ° = 540 ° Example 1. 30 ° is what fraction of a circle, or of one revolution? Answer. 30 ° is   |   30 360|    of a revolution:| 30 360|   Ã‚  =  Ã‚  |   3 3 6|   Ã‚  =  Ã‚  |   1 12| Problem 2. What fraction of a revolution is each of the following? a)    60 °Ã‚     |   60 360|   Ã‚  =  Ã‚  |   6 36|   Ã‚  =  Ã‚  | 1 6| b)    45 °Ã‚     |   45 360|   Ã‚  =  Ã‚  |   5 40|   Ã‚  =  Ã‚  | 1 8| c)    72 °Ã‚     |   72 360|   Ã‚  =  Ã‚  |   8 40|   Ã‚  =  Ã‚  | 1 5| Example 2. If the diameter of a circle is 16 cm, how long is the arc intercepted by a central angle of 45 °? Answer. 45 ° is one eighth of a full circle. It is half of 90  °, which is one quarter. )   Now, the full circumference of this circle is C = ? D = 3. 14 ? 16 cm. The intercepted arc is one eighth of the circumference: 3. 14 ? 16 ? 8 = 3. 14 ? 2   =   6. 28 cm Problem 3. If the diameter of a circle is 20 in, how long is the arc intercepted by a central angle of 72 °? We saw in Problem 2c) that 72 ° is one fifth of a circle. The circumference of this circle is   C = ? D = 3. 14 ? 20 in. The interc epted arc is one fifth of this:   3. 14 ? 20 ? 5 = 3. 14 ? 4 = 12. 56  in. The four quadrants The x-y plane is divided into four quadrants. The angle begins in its standard position in the first quadrant ( I ). As the angle continues in the counter-clockwise direction we name each succeeding quadrant. Why do we name the quadrants in the counter clockwise direction? Because in what we call the first quadrant, the algebraic signs of x and y are positive. Problem 4. In which quadrant does each angle terminate? a)    15 °Ã‚  Ã‚   I     Ã‚     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   b)    ? 15 °Ã‚  Ã‚   IV     Ã‚     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   c)    135 °Ã‚  Ã‚   II     Ã‚   d)    390 °Ã‚  Ã‚   I. 390 ° = 360 ° + 30 °Ã‚     Ã‚     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   e)    ? 100 °Ã‚  Ã‚   III     Ã‚   f)    ? 460 °Ã‚  Ã‚   III. ?460 ° = ? 360 ° ? 100 °Ã‚     Ã‚     Ã‚  g)    710 °Ã‚  Ã‚  | IV. 10 ° is 10 ° less than two revolutions, which are 720 °. | Coterminal angles Angles are coterminal if, when in the standard position, they have the same terminal side. For example, 30 ° is coterminal with 360 ° + 30 ° = 390 °. They have the same terminal side. That is, their terminal sides are indistinguishable. Any angle ? is coterminal with ? + 360 ° because we are just going around the circle one complete time. ?90 ° is coterminal with 270 °. Again, they have the same terminal side. Notice:    90 ° plus 270 ° = 360 °. The sum of the absolute values of those coterminal angles completes the circle. Problem 5. Name the non-negative angle that is coterminal with each of these, and is less than 360 °. a)    360 °Ã‚  Ã‚   0 °Ã‚  Ã‚  Ã‚     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   b)    450 °Ã‚  Ã‚   90 °. 450 ° = 360 ° + 90 °Ã‚   c)    ? 20 °Ã‚  Ã‚   340 °Ã‚  Ã‚  Ã‚     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   d)    ? 180 °Ã‚  Ã‚   +180 °   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   e)    ? 270 °Ã‚  Ã‚   90 ° f)    720 °Ã‚  Ã‚   0 °. 720 ° = 2 ? 360 ° g)    ? 200 °Ã‚  Ã‚  | 160 °| The Radian Measure THE RADIAN SYSTEM of angular measurement, the measure of one revolution is 2?. (In the next Topic, Arc Length, we will see the actual definition of radian measure. ) Half a circle, then, is ?. And, most important, each right angle is half   Ã‚  of ? :   | ? 2| . | Three right angles will be   3 ·   | ? |    =  | 3? 2  | . | Five right angles will be  Ã‚  | 5? 2  | . And so on. | Radians into degrees The following radian measures come up frequently, and the stu dent should know their degree equivalents: ? 4|    is half of   | ? 2|   , a right angle, and so it is equal to 45 °. | Equivalently,  | ? 4|    is of one quarter of ? , or half of half of 180 °. | ? 3|   is a third of ? , and so is equal to 180 ° ? 3 = 60 °. | ? 6|   is a sixth of ? , and so is equal to 180 ° ? 6 = 30 °. | 5? 4  |   Ã‚  =  Ã‚  5 ·Ã‚  | ? 4  |   Ã‚  =  Ã‚  5 ·Ã‚  45 ° = 225 °. | 2? 3  |   Ã‚  is a third of 2?. A third of a revolution = 360 ° ? 3 = 120 °. | Problem 1 . Convert each of these radian measures into degrees. Problem 1. The student should know these. To see the answer, pass your mouse over the colored area. To cover the answer again, click Refresh (Reload). a)    ? 180 °Ã‚  Ã‚  Ã‚  Ã‚  |   Ã‚   b)   Ã‚  | ? 2|   Ã‚   90 °Ã‚  Ã‚  Ã‚  Ã‚   |   Ã‚   c)   Ã‚  | ? 3|   Ã‚   60 °Ã‚  Ã‚  Ã‚  Ã‚   |   Ã‚   d)   Ã‚  | ? 6|   Ã‚   30 °Ã‚  Ã‚  Ã‚  Ã‚   |   Ã‚   e)   Ã‚  | ? 4|   Ã‚   45 ° | Problem 2. Convert each of these radian measures into degrees. a)      Ã‚  | ? 8|   Ã‚  Ã‚  | 22?  °. | ? 8|   is half of  | ? 4| . | b)   Ã‚  | 2? 5|   Ã‚  Ã‚  | 72 °. | 2? 5 |   is a fifth of 2? | ,  which is a fifth of a 360 °. | c)   Ã‚  | 7? 4|   |   = 7 ·Ã‚  | ? 4 |   = 7 · 45 ° = 315 °| d)      | 9? 2|   |   = 9 ·Ã‚  | ? |   = 9 · 90 ° = 810 °| e)      | 4? 3|   |   = 4 ·Ã‚  | ? 3 |   = 4 · 60 ° = 240 °| f)      | 5? 6|   |   = 5 ·Ã‚   | ? 6 |   = 5 · 30 ° = 150 °| g)   Ã‚  | 7? 9|   | | Problem 3. Evaluate the following. a)   cos  | ? 6|   =  | 2|   |   Ã‚  b)   sin  | ? 6|   =  | 1 2|   |   Ã‚  c)   tan  | ? 4|   =  | 1| | d)   cot  | ? 3|   =  |   1 |   |   Ã‚  e)   csc  | ? 6|   =  | 2  |   |   Ã‚  f)   sec  | ? 4|   =  | | Problem 4. In terms of radians, what angle is the complement of an angle  ? ?     | ? 2|   ? | ? | Problem 5. A function of any angle is equal to the cofunction of its complement. Therefore, in terms of cofunctions:   Ã‚  a)   sin  ? =  | cos  (| ? 2|   ? | ? | )|   |   Ã‚  b)   cot  ?   | tan  (| ? 2|   ? | ? | )| c)   sec  (| ? 2|   ? ?)|   =  csc ? | Degrees into radians 360 °   =   2?. When we write 2? , we mean 2? radians, which is approximately 6. 28 radians. However, we normally omit the word radians. As we will see in the next Topic, Arc length, the radia n measure can be any real number. Problem 6. The student should begin by knowing these. 0 °Ã‚  Ã‚  =  | 0 radians. |   | 360 °Ã‚  =  | 2?. |   | 180 °Ã‚  =  | ?. |   | 90 °Ã‚  =  | ? 2| . | 45 °Ã‚  =  | ? 4| . |   | 60 °Ã‚  =  | ? 3| . |   | 30 °Ã‚  =  | ? 6| . | Example 1. Convert 120 ° into radians. Solution. We can go from what we know to what we do not know. In the most important cases we can recognize the number of degrees as a multiple of 90 °, or 45 °, or 60 °, or 30 °; or as a part of 360 °. Since 60 ° =  | ? 3| , then| 120 ° = 2 ·Ã‚  60 ° = 2 ·Ã‚  | ? 3|   =  | 2? 3| . | Or, since 120 ° is a third of 360 °, which is 2? , then 120 °   =   | 2? 3| . | Example 2. 225 ° =  180 ° + 45 °   =   ? +  | ? 4|    =   | 5? 4| . | Or, 225 ° = 5 ·Ã‚  45 °   =   5 ·Ã‚  | ? 4|    =   | 5? 4| . | Problem 7. Convert each of the following into radians. a)  Ã‚  270 °Ã‚  =  | 3 ·Ã‚  90 °Ã‚  =   | | 3? 2|   |   Ã‚  b)  Ã‚  210 °Ã‚  =  | 7 ·Ã‚  30 ° =  7 ·Ã‚  | ? 6|   =  | 7? 6| c)  Ã‚  300 °Ã‚  =  | 5 ·Ã‚  60 ° =  5 ·Ã‚  | ? 3|   =  | 5? 3|   |   Ã‚   d)  Ã‚  135 °Ã‚  =  | 3 ·Ã‚  45 ° =  3 ·Ã‚  | ? |   =  | 3? 4| e)  Ã‚  720 °Ã‚  =  |   2 · 360 ° = 2 · 2? = 4? | f)  Ã‚  450 °Ã‚  =  |   5 · 90 ° = 5 ·Ã‚  | ? 2|   =  | 5? 2| g)  Ã‚  36 °Ã‚  Ã‚  =  | A tenth of 360 °Ã‚  =  | 2? 10|   =  | ? 5| h)  Ã‚  72 °Ã‚  Ã‚  =  | 2 ·Ã‚  36 ° =  | 2? 5| 72 ° is thus a fifth of a revolution. i)  Ã‚  40 °Ã‚  Ã‚  =  | A ninth of 360 °Ã‚  =  | 2? 9| j)  Ã‚  80 °Ã‚  Ã‚  =  | 2 ·Ã‚  40 ° =  | 4? 9| As a last resort, proportionally, so that Example 3. Change 140 ° to radians. Solution. | 140 180|  ·Ã‚  ? |    =   | 7 9|  ·Ã‚  ? |    =   | 7? 9| ,| upon dividing both the numerator and denominator first by 10 and then by 2Coterminal angles Angles are coterminal if they have the same terminal side. ? is coterminal with . They have the same terminal side. Notice that ? + ? =   2? , so that ?   =   2? ? ? .   . .   . .   . .   . (1) Example 4. Name in radians the non-negative angle that is coterminal      with  ? | 2? 5| , and is less than 2?. | Answer. Let us call that angle ?. Then according to line (1), ? =   2? ?  | 2? 5|   Ã‚  =  Ã‚  | 10? ? 2? 5|   Ã‚  =  Ã‚  | 8? 5| Problem 8. Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2?. a)  Ã‚  ? | ? 6| . | ? =   2? ?  | ? 6|   Ã‚  =  Ã‚  | 12? ? ? 6|   Ã‚  =  Ã‚  | 11? 6| b)  Ã‚  ? | 3? 4| . | ? =   2? ?  | 3? 4|   Ã‚  =  Ã‚  | 8? ? 3? 4|   Ã‚  =  Ã‚  | 5? 4| c)  Ã‚  ? | 4? 3| . | ? =   2?   | 4? 3|   Ã‚  =  Ã‚  | 6? ? 4? 3|   Ã‚  =  Ã‚  | 2? 3| The multiples of ? Starting at 0, let us go around the circle a half-circle at a time. We will then have the following sequence, which are the multiples of ? : 0,   ? ,   2? ,    3? ,   4? , 5? , etc. The point to see is that the odd multiples of ? , ?,   3? ,   5? ,   7? , etc. are coterminal with ?. While the even multiples of ? , 2? ,   4? ,   6? , etc. are coterminal with 0. If we go around in the negative direction, we can make a similar observation. Problem 9. Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2?. )    -? ?     Ã‚  Ã‚  Ã‚   b)    -2? 0     Ã‚  Ã‚  Ã‚   c)    -3? ?     Ã‚  Ã‚  Ã‚   d)    -4? 0     Ã‚  Ã‚  Ã‚   e)    -5? ?   f)    3? ?     Ã‚  Ã‚      g)    4? 0     Ã‚  Ã‚      h)    5? ?     Ã‚  Ã‚      i)    6? 0     Ã‚     Ã‚   j)    7? ? IT IS CONVENTIONAL to let the letter s symbolize the length of an arc, which is called arc length. We say in geometry that an arc subtends an angle ? ; literally, stretches under. Now the circumference of a circle is an arc length. And the ratio of the circumference to the di ameter is the basis of radian measure. That ratio is the definition of  ?. ?|    =   | C D| . | Since D = 2r, then ?| =| C 2r| or, C r|   =   | 2? | . | That ratio 2? of the circumference of a circle to the radius, is called the radian measure of 1  revolution, which are four right angles at the center. The circumference subtends those four right angles. Radian measure of ? =   | s r| Thus the radian measure is based on ratios numbers that are actually found in the circle. The radian measure is a real number that names the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle ?. | Theorem. |   | In any circles the same ratio of arc length to radius|   |   | determines a unique central angle that the arcs subtend. Proportionally, if and only if ?1 = ? 2. We will prove this theorem below. Example 1. If s is 4 cm, and r is 5 cm, then the number  | 4 5| ,  i. e. | s r| ,  is the| radian measure of the central angle. At that central angle, the arc is four fifths of the radius. Example 2. An angle of . 75 radians means that the arc is three fourths of the radius. s = . 75r Example 3. In a circle whose radius is 10 cm, a central angle ? intercepts an arc of 8 cm. a)   What is the radian measure of that angle? Answer. According to the definition: ?   =   | s r|    =   |   8 10|    =   . 8| b)   At that same central angle ? what is the arc length if the radius is b)   5  cm? Answer. For a given central angle, the ratio of arc to radius is the same. 5 is half of 10. Therefore the arc length will be half of 8:   4cm. Example 4. a)   At a central angle of 2. 35 radians, what ratio has the arc to the radius? Answer. That number is the ratio. The arc is 2. 35 times the radius. b)   In which quadrant of the circle does 2. 35 radians fall? Answer. Since ? = 3. 14, then  | ? 2|   is half of that:   1. 57. | 3? 2|    = 3. 14 + 1. 57| = 4. 71. An angle of 2. 35 radians, then, is greater than 1. 57 but less that 3. 14. It falls in the second quadrant. = r? c)   If the radius is 10 cm, and the central angle is 2. 35 radians, then how c)   long is the arc? Answer. We let the definition of ? , ?   =   | s r| become a formula for finding s : s   =   r? | Therefore, s   =  10 ? 2. 35 = 23. 5 cm Because of the simplicity of that formula, radian measure is used exclusively in theoretical mathematics. The unit circle Since in any circle the same ratio of arc to radius determines a unique central angle, then for theoretical work we often use the unit circle, which is a circle of radius 1:   r = 1. In the unit circle, the length of the arc s is equal to the radian measure. The length of that arc is a real number x. s = r? = 1 ·Ã‚  x = x. We can identify radian measure, then, as the length x of an arc of the unit circle. And it is here that the term trigonometric function has its full meaning. For, corresponding to each real number x each radian measure, each arc there is a unique value of sin x, of cos x, and so on. The definition of a function is satisfied. (Topic 3 of Precalculus. ) Moreover, when we draw the graph of y = sin x (Topic 18), we can imagine the unit circle rolled out in both directions onto the x-axis, and in that way marking the coordinates ? , 2? , , ? 2? and so on, on the x-axis. Because radian measure can be identified as an arc, the inverse trigonometric functions have their names. arcsin is the arc the radian measure whose sine is a certain number. The ratio  | sin x x| In the unit circle, the opposite side AB is sin x. sin x| =| AB 1| =   AB. | One of the main theorems in calculus concerns the ratio  | sin x   Ã‚     x|   for| very small values of x. And we can see that when the point A on the circumference is very close to C that is, when the central angle AOC is very, very small then the opposite side AB will be virtually indistinguishable from the arc length AC. That is, sin x| | x| | sin x x| | 1. | An angle of 1 radian An angle of 1 radian refers to a central angle whose subtending arc is equal in length to the radius. That is often cited as the definition of radian measure. Yet it remains to be proved that if an arc is equal to the radius in one circle, it will subtend the same central angle as an arc equal to the radius in another circle. We cannot avoid the main theorem. In addition, although it is possible to define an angle of 1 radian, does such an angle actually exist? Is it possible to draw one a curved line equal to a straight line? Or is that but another example of fantasy mathematics? See First Principles of Euclids Elements, Commentary on the Definitions; see in particular that a definition asserts only how a word or a name will be used. It does not assert that what has been defined exists. Problem 1. a)   At a central angle of   | ? 5| ,  approximately what ratio has the arc to the| a)   radius? Take ? 3. The radian measure  | ? 5|   is that ratio| . Taking ? 3, then the| arc is approximately three fifths of the radius. b)   If the radius is 15 cm, approximately how long is the arc? s = r? 15 ·Ã‚  | 3 5|   = 9 cm| Problem 2. In a circle whose radius is 4 cm, find the arc length intercepted by each of these angles. Again, take ? 3. a)  Ã‚  | ? 4|   Ã‚  Ã‚  | s = r? 4 ·Ã‚  | 3 4|   = 3 cm| b)  Ã‚  | ? 6|   Ã‚  Ã‚  | s = r? 4 ·Ã‚  | 3 6|   = 4 · ? = 2 cm| c)  Ã‚  | 3? 2|   Ã‚  Ã‚  | s = r? 4 ·Ã‚  | 3 · 3 2|   = 4 ·Ã‚  | 9 2|   = 2 · 9 = 18 cm| d)  Ã‚  2?. (Here, the arc length is the entire circumference! ) s = r? = 4 ·Ã‚  2? 4 ·Ã‚  6 = 24 cm| Problem 3. In which quadrant of the circle does each angle, measured in radians, fall? (See the figure above. )      a)   ? = 2|   Ã‚  Ã‚  | 2 radians are more than  | ? 2|   but less than ?. (See the| figure above. )   Therefore, ? 2 falls in the second quadrant. b)   ? = 5|   Ã‚  Ã‚  | 5 radians are more than  | 3? 2|   but less than 2?. (See the| figure above. )   Therefore, ? = 5 falls in the fourth quadrant. c)   ? = 14|   Ã‚  Ã‚  | 14 radians are more than 2 revolutions, but slightly| less than 2? :   6. 28 + 6. 28 = 12. 56. (See the figure above. ) Therefore, ? = 14 falls in the first quadrant. Proof of the theorem In any circles the same ratio of arc length to radius determines a unique central angle that the arcs subtend; and conversely, equal central angles determine the same ratio of arc length to radius. Proportionally, if and only if ?1 = ? 2. For, if and only if Now 2? r is the circumference of each circle. And each circumference is an arc that subtends four right angles at the center. But in the same circle, arcs have the same ratio to one another as the central angles they subtend. Therefore, and Therefore, according to line (1), if and only if ?1 = ? 2. Therefore, the same ratio of arc length to radius determines a unique central angle that the arcs subtend. Basic Concepts: In Brief, The Sexagesimal System, Centesimal System and the radian measure help in converting the angles.