Exam 3: CAR100 Angles 2023-2024 PDF

Exam 3 Who When 𝑥|𝑥 𝑖𝑠 𝑎 𝑠𝑡𝑢𝑑𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑖𝑠 𝑐𝑙𝑎𝑠𝑠 Nov 1, 8:30 PM What Time limit: 90 minutes Lectures 19 – 27 23 multiple choice, 2 free response Where CAR100...

Exam 3 Who When 𝑥|𝑥 𝑖𝑠 𝑎 𝑠𝑡𝑢𝑑𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑖𝑠 𝑐𝑙𝑎𝑠𝑠 Nov 1, 8:30 PM What Time limit: 90 minutes Lectures 19 – 27 23 multiple choice, 2 free response Where CAR100 Valentino, Howladar, Bridges, Jauhari NRN1020 Beye, Craddock Bring #2 pencil and a physical picture ID only Preparation: Quiz 8, Old exams, Review class questions Jangles  Angles 𝑒𝜋 163 is almost an integer What’s an angle? Angle Basics An angle is formed from two rays that have a common endpoint. The rays are initially on top of each other, and one ray is rotated around the endpoint. The starting position is the initial side. The ending position is the terminal side. The endpoint of the ray is the vertex. Often named using Greek letters: 𝜃, 𝜙, 𝛼, 𝛽 Standard Position and Orientation An angle is in standard position if its initial side is on the positive 𝑥-axis, and its vertex is at the origin. A rotation in the counterclockwise direction is positive. A rotation in the clockwise direction is negative. An angle can be more than a full rotation Why do clocks go clockwise? Central Angle A central angle is an angle whose vertex is at the center of a circle If the angle is also in standard position then it is a central angle in standard position Measuring Angles One common unit for measuring angles is the degree 1 An angle of 1° is of a full rotation 360 i.e. one full rotation is an angle of Why 360? Thank the Babylonians, Egyptians, and Sumerians 360 works nicely in a system based around the number 60 360 is nicely divisible by many numbers 360 is approximately the number of days in a solar year Sure, but… A More Civilized Angle Measure To define a new way to measure angles, we need to define what a measure of 1 will represent 1 Recall: 1° is of a full rotation 360 Look at the relationship between central angle and arc length Radians An angle of 1 radian is defined as the central angle that creates an arc whose length is the same as the radius Arc Length and Radians What is the arc length created by an angle of… 2 radians? 𝑠= 3 radians? 𝑠= In general, the arc length created by an angle of 𝜃 radians is… 𝑠= How many radians in one full rotation? One full rotation corresponds to an arc length of Axis Angles Other important angles in radians Coterminal Angles Two angles are coterminal if they have the same initial and terminal sides If 𝜃 is measured in radians then any angle that is coterminal with 𝜃 has the form 𝜋 Find 3 angles that are coterminal with 2 Supplementary Angles Two angles 𝜃1 and 𝜃2 are Examples supplementary if their sum is a line 𝜃1 = 40° 𝜃2 = In radians: 𝜃1 + 𝜃2 = In degrees: 𝜃1 + 𝜃2 = 𝜋 𝜃1 = 𝜃2 = 5 Complementary Angles Two angles 𝜃1 and 𝜃2 are Example: complementary if their sum is a 3𝜋 right angle 𝜃1 = 7 In radians: 𝜃1 + 𝜃2 = In degrees: 𝜃1 + 𝜃2 = 𝜃2 = Converting Between Degrees and Radians Compare a full rotation in each Examples system 𝜃 = 120° In radians: In degrees: Conversion Factor 7𝜋 𝜃= 3 Sector Area Area of entire circle: A sector is a region inside a circle created by a central angle What is the area of a sector enclosed by an angle 𝜃? 𝐴= Sector Area Example A lawn sprinkler rotates through shoots water up to 20 feet. It needs to cover an area of 800 square feet. What angle does it need to rotate through to achieve this? Rotation and Movement When an object is moving in a circular path, there are two ways to describe its velocity Linear speed – how far the object travels per unit of time 𝑣= Angular speed – how many rotations the object completes per unit of time 𝜔= How are they related? Rotation and Movement Examples An LP vinyl record has a 12-inch diameter and rotates 100 times every 3 minutes. What is the linear speed (in inches per second) of the outer edge of the record? Rotation and Movement Examples What is the linear speed in miles per hour of a point on Earth’s equator? Rotation and Movement Examples A circular pulley with a radius of 16 inches is attached to another pulley with a radius of 2 inches. If the larger pulley is spinning at a rate of 3 rotation per second, how fast is the smaller pulley spinning (in rotations per second)? Rotation and Movement Examples A bicycle consists of two circular gears connected by a chain. The rider turns the front gear and the bicycle tire is connected to the rear gear. How are the linear and angular velocities of the front gear F, rear gear R, and rear tire T related? Rotation and Movement Examples Suppose a bike’s front gear has a radius of 3.3 inches, rear gear has a radius of 1.1 inches, and the tire has a diameter of 26 inches. If the rider pedals the front gear at 60 rpm how fast will the bicycle go?