1. 4.1 Day 2 Objectives: Find coterminal angles Find the length of a circular arc Use linear & angular speed to describe motion on a circular path Pg. 459 #58-82 (evens), 88, 90, 92, 98 Helpful Hints: For #76, see the example regarding hard drives on Pg. 457 For #90 and #92, see the arc length example in this packet. For #98, see the example on the last page of the packet.

2. Coterminal Angles Coterminal Angles have rotating rays that stop at the same position on the circle. This kind of ray is called the “terminal side” of the angle. Angle may be positive or negative (move counterclockwise or clockwise). A 70 degree angle is coterminal to a -290 degree angle, for example. Angle may go around the circle more than once. This means a 30 degree angle is coterminal to a 390 degree angle.

3. Coterminal Angles Angles are coterminal if they have the same terminal side.coterminal θ is coterminal with −φ. They have the same terminal side To find a positive angle less than 2π that is coterminal with a given angle: If the given angle is positive, subtract 2π (or 360 o ) as many times as necessary. If the given angle is negative, add 2π (or 360o) as many times as necessary.

4. Find a positive angle less than 360 o that is coterminal with each of the following: a)400 o b) -135 o c) 855 o Find a positive angle less than 2π that is coterminal with each of the following: d) e) f) g)

5. Arc Length Recall theta Θ (in radians) is the ratio of arc length to radius or Therefore, Arc length = radius x theta (in radians) Example: A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45 o. (Express the length in terms of π).

6. Linear speed & Angular speed Speed a particle moves along an arc of the circle (v) is the linear speed (distance, s, per unit time, t) Speed which the angle is changing as a particle moves along an arc of the circle is the angular speed. (angle measure in radians, per unit time, t) Linear speed in terms of angular speed :

7. Relationship between linear speed & angular speed Linear speed is the product of radius and angular speed. Example: The minute hand of a clock is 6 inches long. How fast is the tip of the hand moving? We know angular speed = 2 pi per 60 minutes

8. Example: Long before iPods that hold thousands of songs and play them with superb audio quality, individual songs were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center.