welcome back mathematicians in this video we are going to discuss angles and we're going to start with the definition of an angle if two rays are drawn with a common vertex they form an angle one ray of the angle is called the initial side and the other ray is called the terminal side so i'm going to start by making my common vertex and i'm going to make my initial side horizontal for this first angle i'm then going to draw my terminal side diagonal and i'm going to show the angle i'm going to identify the angle by showing the direction and amount of rotation from the initial side to the terminal side so i will do this with a curved arrow so i'm going to start with the initial side and draw my curved arrow towards the terminal side if the rotation is counterclock in the counterclockwise direction the angle is positive because this starts at the initial side and has a counterclockwise direction this is a positive angle now what if i draw another angle my initial side is still horizontal i'm going to call that is for short and my terminal side is also again diagonal i'm going to call that ts for short but in this case i'm going to change the rotation i'm going to start with my initial side but i'm going to rotate clockwise around the coordinate plane around the angle to the terminal side and if the rotation is clockwise in the clockwise direction so if again that curved arrow is in the clockwise direction the angle is negative so this is a negative angle next let's define standard position an angle theta and that is the greek letter theta is said to be in the standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis i'm going to go ahead and draw two angles in the first coordinate plane i'm going to put the vertex on the origin the initial side will coincide with the positive x-axis and i'm going to put the terminal side in quadrant number one and so what i'm then going to do is draw an angle and i'll make this angle counterclockwise direction which means that this angle theta is positive i'm going to draw a second angle the vertex is still on the origin the initial side is still coinciding with the positive x-axis in this case i'm going to make the terminal side in quadrant number 3. i'm going to draw the curved arrow to indicate rotation but i'm going to go in the clockwise direction and in this case that indicates that theta is negative both of these represent angles that are in standard position as long as the vertex is on the origin and the initial side is coinciding with the positive x-axis now what i'd like to do is draw each of these angles and this these are really just sketches i want to be clear in order to do these accurately i would need tools or i would need technology so i'm just sketching these i'm going to draw each angle in standard position and i'm going to start with 30 degrees so the first thing i'm going to do is draw my initial side and my vertex so all of these are going to have the same initial side and the same vertex when drawn in standard position next i'm going to identify what each of my axes represent in terms of degrees so the positive x-axis represents 0 degrees the positive y axis represents a 90 degree angle then if we rotate further to the negative x axis that actually represents a positive 180 degree angle then if we rotate to the negative y axis that represents a positive 270 degrees and then finally if we continue to rotate all the way back around to the positive x-axis that also represents a positive 360 degrees so i now that i know this i can tell you that the 30 degree angle will be in quadrant number one with a counterclockwise rotation and it is important to to identify the rotation with that curved arrow so in this case again counterclockwise now is that exactly where it's located in the coordinate plane probably not but this is just a sketch next i'm going to do negative 120 degrees so i will start once again with the vertex and the initial side but in this case because of the fact that it's negative 120 degrees i know i'm going to go clockwise now some of you may be able to do it using the angles that i marked in the first problem but what you could do is actually rotate the other way around marking the axes accordingly so yes negative y-axis is 270 degrees but it's also negative 90 degrees and then that means that the negative x-axis is negative 180 degrees positive y-axis would be negative 270 degrees and then zero degrees would also be negative 360 degrees if we continue to rotate given that information i know that negative 120 degrees will be in quadrant number three because negative 120 is between negative 90 and negative 180 keeping in mind i do need to identify the rotation so i will start with the initial side and rotate clockwise to the terminal side next i'm going to go to 450 degrees now what hopefully you recognize right away with 450 degrees is that 450 degrees is larger than 360. so we will start once again with the vertex and the initial side i'm going to again mark the positive x-axis as 0 degrees then we have 90 degrees then we have 180 degrees then we have 270 degrees we also have 360 for the positive x-axis i can keep going if i want to to the positive y-axis add another 90 degrees and this is 450 degrees so this tells me that my terminal side actually should be on the positive y-axis now what i should do is number one identify the rotation which because it is a positive angle it will be counterclockwise but because it actually is 450 degrees which is one full revolution plus some extra i need to identify that so i'm going to start on the initial side and i'm going to make my arrow go all the way around the coordinate plane to identify that it went around 360 degrees and it continues further until it ends at 450 degrees so that's how you would mark an angle that is larger than 360 degrees you would do the same if it was negative 450 degrees but you would then go clockwise direction instead of counterclockwise direction and maybe marking your angles like we did in the second problem all right guys i hope that helps thanks