Class Note for MATH 111 at UA
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.
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Date Created: 02/06/15
11 Angles Line Let A and B be two distinct points We can draw a unique line passing through A and B and we will call it line AB By this we mean a set of points that stretches from in nity on one side passes through A then through B and goes on to in nity on the other side Ray If we drop the part of the line AB that lies before the point A what remains is called ray AB Thus ray AB is the portion of the line AB that starts at A continues through B and on past B to in nity Here point A is called the endpoint of ray AB We will now use the concept of a ray to de ne the notion of angle Angle An angle is formed by rotating a ray around its endpoint The ray in its initial position is called the initial side of the angle while the ray in its location after the rotation is the terminal side of the angle The endpoint of the ray is the vertex of the angle Note the close relation between the notions of angle and rotation in the de nition above There is more to an angle than just a vertex and two rays with endpoints located in the vertex Given a vertex an initial side and a terminal side we have not de ned an angle in a unique way there are obviously two different angles sharing the same initial and terminal sides 7 one by rotating a ray from the initial side to the terminal side clockwise and one by performing a rotation from the initial side to the terminal side counterclockwise1 Therefore the key word in the de nition of an angle is rotation Positive and Negative Angles As we identi ed in a way an angle with a rotation and as we distinguish two kinds of rotations clockwise and coun terclockwise we can use this property to classify angles into two categories accordingly If an angle is formed by a counterclockwise rotation we will say that the angle is positive If an angle is formed by a clockwise rotation the angle will be said to be negative Degree Measure The measure of angles allows us to compare angles that do not share the same vertex and initial side One of the most common units for measuring angles is the degree It is de ned by assigning a numerical value of 360 to the angle that corresponds to a counterclockwise full rotation Al ternatively it could be said that an angle of 360 is the smallest positive angle such that its initial and terminal sides coincide Right Angle7 Straight Angle Having chosen a measure for an angle cor responding to a full rotation we can nd measures for angles that are fractions thereof 1In fact there are in nitely many angles for given initial and terminal sides 7 see the paragraph on coterminal angles below An angle obtained by rotating a ray by a half of a full rotation has a measure of gtlt 360 180 and is called a straight angle An angle of 90 corresponds to a quarter of a full rotation and is called a right angle Acute Angles Obtuse Angles Angles measuring between 0 and 90 are called acute angles Angles measuring between 90 and 180 are called obtuse angles Complementary and Supplementary Angles Two positive angles the sum of which is 90 are called complementary If the sum of two positive angles is 180 they are said to be supplementary Note that if two angles are complementary they are necessarily both acute If two angles are supplementary they are either both right or one is acute and the other obtuse Standard Position For reference purposes it is convenient to introduce co ordinate axes z and y An angle is said to be in standard position if its vertex is located at the origin and its initial side is along the positive zaxis An angle in standard position is said to lie in the quadrant in which its terminal side lies An acute angle lies in quadrant l and an obtuse angle lies in quadrant 11 In standard position a right angle has its terminal side along the positive y axis while a straight angle has its terminal side along the negative zaxis These two are examples of quadrantal anglesiangles that have their terminal side along any coordinate axis Coterminal Angles So far we have only seen angles corresponding to a full rotation or less but there is nothing preventing the rotation from going on after a full rotation in that way we get angles measuring more than 360 Obviously the terminal side of any angle with a measure greater than 360 coincides with the terminal side of some angle less than 360 This is the case with any two angles differing by a full rotation or multiple thereof Such angles are called cotermz39nal angles th could be said that these are angles that lie in between quadrants or which delineate quadrants
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