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MAC1114 Lecture Note Outlines_Eichler Lecture 1 (L1): Angles and Angle Measures Textbook Section: 4.1 Geometry Review: Point Line segments Ray Vertex Cartesian plane Angles: An __________________________ is formed by rotating a _______________ about its endpoint or ________________________. The starting position is called the ____________________________ of the angle and the ending position is called the ______________________________. Angles in the Coordinate Plane: An angle is said to be in __________________________________ if its initial side is located on the ________________________________________ , its vertex at the _______________________ , and has been rotated _______________________________ about its vertex. _________________________ angles have the same initial side and vertex position, but have been rotated____________________________________. Notation and examples: 1 MAC1114 Lecture Note Outlines_Eichler Angle Measure: Angles are measured by the amount of _____________________ from initial to terminal side. When measuring angles on the coordinate plane it is convenient to imagine the angle sitting inside of a _______________________ because if you rotate an angle in the coordinate plane completely around its vertex, the path the angle makes forms a _____________________. A _________________________ is defined as _____________ of a complete rotation. A complete rotation measures_______________________, a half rotation measures______________________, a quarter rotation measures__________________________________, and three-quarters rotation measures____________________________. Diagram: Another way of measuring angles is using ____________________________. One___________________ is the measure of a central angle 𝜃 that intercepts __________________________________that has length equal to _______________________________________. Diagram and algebraic representation: Problem: How many radians is a full revolution? How many radians is a half, a quarter, and three-quarters revolution? 2 MAC1114 Lecture Note Outlines_Eichler Problem: How do we convert between different angle measures? Angle pairs: Two positive angles are said to be ___________________________________ if their sum is ________________ or ________________. Two angles positive are said to be ___________________________ if their sum is _________________ or _____________________. Problem: Find the supplement and complement of 2𝜋 3 . Angles that have the same ______________________ and ________________________ sides are said to be _____________________________________. Problem: Find two angles, one positive and one negative, that are conterminal with 3 2𝜋 3 . MAC1114 Lecture Note Outlines_Eichler Arc Length: Recall that the radian measure of an angle 𝜃 is given by ________________________________ where _____ represents the ________________________________________________________ and ______ is the _________________________________________________. Problem: How can we use the formula for radian measure to find a formula for the arc length of a circle? Problem: Find the length of the arc of a circle with radius 3 intercepted by a central angle of 140° . NYTI Problem: Given a circle with radius 20, find the length of the arc intercepted by a central angle of 250° . 4 MAC1114 Lecture Note Outlines_Eichler Problem: Say a particle is moving at a constant speed along a circular arc of radius 2. If the particle travels an arc length of 4 units in 3 seconds, how fast is the particle moving along the circle? Linear and Angular speed: The ____________________________________ of a particle moving at a constant rate along a circular arc with a radius of _______ is the ____________________________________________ traveled in _________________ . Diagram and algebraic representation: The __________________________________________ of the particle is found by considering the measure of _______________________ corresponding to the arc length 𝑠 traveled in time 𝑡. Diagram and algebraic representation: 5 MAC1114 Lecture Note Outlines_Eichler Problem: If the minute hand of a clock is 3 inches long, what is the linear and angular speed of the minute hand? Problem: Two pulleys are connected by a moving belt. One pulley has radius 3 cm and the other has radius 9 cm. If the smaller pulley is rotating at 6 revolutions per minute, how many revolutions per minute is the larger pulley rotating? A _________________________ of a circle is a part of the circle that is bounded by ____________________________________________________ and their _______________________________________. Problem: How can we find the area of a sector of a circle given the formula for the area of a circle with radius 𝑟 is 𝐴 = 𝜋𝑟 2 ? Area of a sector of a circle: The area of a sector of a circle 𝐴 with radius 𝑟 and central angle 𝜃, in __________________, is given by ________________________________________________. Problem: On a clock with a minute hand 3 inches long, how much area of the clock face is covered by the minute hand in 15 minutes? 6