Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Angles Angles are made-up of two rays, an initial side and a terminal side. The two rays come together at a point called the vertex. If the angle is going counterclockwise, it's considered a positive angle. If it's going clockwise, it's a negative angle. Standard position is when the initial side is along the X axis positive direction with the vertex at the origin. Then any angle with the initial side on the X axis and the vertex at the origin is considered standard position. Theta lies in a quadrant, or Theta is a quadrant angle. If Theta literally goes into any of the four quadrants, it's a. It lies in a quadrant. However, if Theta lays along one of the axis, it's considered a quadrant angle. Degrees and radians of a circle. 1° equals one 360 revolution of a circle. 90° is a right angle, 180° is a straight angle. One revolution or one time around the circle is 360 degrees. 1° = 60 minutes. One minute equals 60 seconds. A Radian by definition says that if the radius of a circle is the same distance as the arc length, the included angle then is considered one Radian. By this definition, one full revolution around a circle is 2π radians. 2π radians could then be thought of as 360° or π radians is 180°. If we divided each side by π, we'd get one Radian equaling 180° / π. If we had divided each side by 180°, we would have gotten 1° equaling π / 180 radians. The arc length of a sector in a circle. The circumference is the formula circumference equals 2π R. The variable that represents arc length of a sector is S, and we're looking at just a portion of the whole circle. We want this length, so we're going to talk about the Theta over the whole circle of 2π. So Theta out of 2π times the original circumference of 2π R, the 2π's cancel and we get S equal Theta times R If we're in radiance, we can also do this in degrees. The equation just doesn't simplify as nicely. So the arc length is going to equal Theta, the portion of the circle we're looking for. Over the entire circle of 360° times the original circumference of 2π R the two and the 360 will reduce to Theta over 180° * π R. The area of a sector of a circle we would look at as using the original area formula, which is area equal π R-squared. But now we don't want the full circle, we want a portion of it. So in radians we want Theta over 2π. The Pi's cancel and we would get reduced to the area of the sector is 1/2 Theta r ^2 in degrees. We'd have area equal Theta over the hundred 360° * π R-squared and that one doesn't simplify so we'd just get 1 / 360° * π R-squared, Theta Pi R-squared. Linear in angular speed. Average speed by definition is the distance traveled divided by the elapsed time in a circle. We have both linear speed and angular speed. The linear speed would be the distance traveled here divided by the time. Remember the distance is S for the arc length, but S also equaled R times Theta. The angular speed is going to be the angle traveled divided by time. So one is distance traveled and one is the angle traveled. Now there's a direct relationship between linear speed and angular speed. V = s / t by our definitional linear speed, but S was really just R times Theta. If we regroup that, we'd have R times Theta over T Theta over T is our angular speed of Omega, so R times Omega. So V. The linear speed is really just equal to R times Omega, R being the radius. Thank you and have a wonderful day. This is Anna Cox.