1. Copyright © 2007 Pearson Education, Inc. Slide 8-1 5.1 Angles and Arcs Basic Terminology –Two distinct points A and B determine the line AB. –The portion of the line including the points A and B is the line segment AB. –The portion of the line that starts at A and continues through B is called ray AB. –An angle is formed by rotating a ray, the initial side, around its endpoint, the vertex, to a terminal side.

2. Copyright © 2007 Pearson Education, Inc. Slide 8-2 Degree Measure –Developed by the Babylonians around 4000 yrs ago. –Divided the circumference of the circle into 360 parts. One possible reason for this is because there are approximately that number of days in a year. There are 360 ° in one rotation. –An acute angle is an angle between 0 ° and 90 °. –A right angle is an angle that is exactly 90 °. –An obtuse angle is an angle that is greater than 90 ° but less than 180 °. –A straight angle is an angle that is exactly 180 °. 5.1 Degree Measure

3. Copyright © 2007 Pearson Education, Inc. Slide 8-3 5.1 Finding Measures of Complementary and Supplementary Angles If the sum of two positive angles is 90 °, the angles are called complementary. If the sum of two positive angles is 180 °, the angles are called supplementary. ExampleFind the measure of each angle in the given figure. (a) (b) (Supplementary angles) (Complementary angles) Angles are 60 and 30 degrees Angles are 72 and 108 degrees

4. Copyright © 2007 Pearson Education, Inc. Slide 8-4 5.1 Calculating With Degrees, Minutes, and Seconds One minute, written 1', is of a degree. One second, written 1", is of a minute. ExamplePerform the calculation Solution ' Since 75' = 1 ° + 15', the sum is written as 84 ° 15'.

5. Copyright © 2007 Pearson Education, Inc. Slide 8-5 5.1Converting Between Decimal Degrees and Degrees, Minutes, and Seconds Example (a) Convert 74 º 814  to decimal degrees. (b) Convert 34.817 º to degrees, minutes, and seconds. Analytic Solution (a)Since

6. Copyright © 2007 Pearson Education, Inc. Slide 8-6 5.1Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (b) Graphing Calculator Solution

7. Copyright © 2007 Pearson Education, Inc. Slide 8-7 5.1 Coterminal Angles Quadrantal Angles are angles in standard position (vertex at the origin and initial side along the positive x- axis) with terminal sides along the x or y axis, i.e. 90 °, 180 °, 270 °, etc. Coterminal Angles are angles that have the same initial side and the same terminal side.

8. Copyright © 2007 Pearson Education, Inc. Slide 8-8 5.1 Finding Measures of Coterminal Angles ExampleFind the angles of smallest possible positive measure coterminal with each angle. (a) 908 ° (b) –75 ° SolutionAdd or subtract 360 ° as many times as needed to get an angle between 0 ° and 360 °. (a) 908 – 2(360) = 188 degrees (b) 360 + (-75) = 285 degrees Let n be an integer, we have an infinite number of coterminal angles: e.g. 60 ° + n· 360 °.

9. Copyright © 2007 Pearson Education, Inc. Slide 8-9 5.1Radian Measure The radian is a real number, where the degree is a unit of measurement. The circumference of a circle, given by C = 2  r, where r is the radius of the circle, shows that an angle of 360 º has measure 2  radians. An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian.

10. Copyright © 2007 Pearson Education, Inc. Slide 8-10 5.1Converting Between Degrees and Radians Multiply a radian measure by 180 º /  and simplify to convert to degrees. For example, Multiply a degree measure by  /180 º and simplify to convert to radians. For example,

11. Copyright © 2007 Pearson Education, Inc. Slide 8-11 5.1Converting Between Degrees and Radians With the Graphing Calculator ExampleConvert 249.8 º to radians. Solution Put the calculator in radian mode. ExampleConvert 4.25 radians to degrees. Solution Put the calculator in degree mode.

12. Copyright © 2007 Pearson Education, Inc. Slide 8-12 5.1Equivalent Angle Measures in Degrees and Radians Figure 18 pg 9

13. Copyright © 2007 Pearson Education, Inc. Slide 8-13 5.1Arc Length ExampleA circle has a radius of 25 inches. Find the length of an arc intercepted by a central angle of 45 º. Solution The length s of the arc intercepted on a circle of radius r by a central angle of measure  radians is given by the product of the radius and the radian measure of the angle, or s = r ,  in radians

14. Copyright © 2007 Pearson Education, Inc. Slide 8-14 5.1Linear and Angular Speed Angular speed  (omega) measures the speed of rotation (angle generated in one unit of time) and is defined by Linear speed (the distance travelled per unit of time)  is defined by Since the distance traveled along a circle is given by the arc length s, we can rewrite  as

15. Copyright © 2007 Pearson Education, Inc. Slide 8-15 5.1Finding Linear Speed and Distance Traveled by a Satellite ExampleA satellite traveling in a circular orbit 1600 km above the surface of the Earth takes two hours to complete an orbit. The radius of the Earth is 6400 km. (a)Find the linear speed of the satellite. (b)Find the distance traveled in 4.5 hours. Figure 24 pg 12

16. Copyright © 2007 Pearson Education, Inc. Slide 8-16 5.1Finding Linear Speed and Distance Traveled by a Satellite Solution (a) The distance from the Earth’s center is r = 1600 + 6400 = 8000 km. For one orbit,  = 2 , so s = r  = 8000(2  ) km. With t = 2 hours, we have (b) s =  t = 8000  (4.5)  110,000 km

17. Copyright © 2007 Pearson Education, Inc. Slide 8-17 Area of a Circular Sector A = must be in radians