1. 1 What you will learn  What radian measure is  How to change from radian measure to degree measure and vice versa  How to find the length of an arc given the measure of the central angle  How to find the area of a sector

2. Objective: 6-1 Angles and Radian Measure 2 Radian Measure In most areas involving application of trigonometry, we use degree measures. In more advanced math work (and many sciences) radian measures are used. It allows real numbers to be used rather than degree measures.

3. Objective: 6-1 Angles and Radian Measure 3 Radians? 1 What is the circumference of a circle with a radius of 1?

4. Objective: 6-1 Angles and Radian Measure 4 More Measurements

5. Objective: 6-1 Angles and Radian Measure 5 Converting to Radians from Degrees We have seen that: Therefore, to change from degrees to radians, multiply the degree measure by:

6. Objective: 6-1 Angles and Radian Measure 6 Example  Convert each degree measure to radians: A. 45 o B. 115 o

7. Objective: 6-1 Angles and Radian Measure 7 Converting Radians to Degrees  To convert radians to degrees:  In other words, multiply the radian measure by:

8. Objective: 6-1 Angles and Radian Measure 8 Example  Convert each radian measure to degrees. A. B.

9. Objective: 6-1 Angles and Radian Measure 9 Finding Function Values for Angles in Radians  Our “special” angles:

10. Objective: 6-1 Angles and Radian Measure 10 Evaluating Angles Measured in Radians  Evaluate  You Try: Evaluate

11. Objective: 6-1 Angles and Radian Measure 11 Circular Arcs and Central Angles  Radian measure can be used to find the length of a circular arc. A circular arc is a part of a circle. The arc is often defined by the central angle that intercepts it.

12. Objective: 6-1 Angles and Radian Measure 12 Finding Arc Length  The length of any circular arc s is equal to the product of the measure of the radius of the circle r and the radian measure of the central angle that it subtends.

13. Objective: 6-1 Angles and Radian Measure 13 An Example  Given a central angle of 128 o, find the length of the intercepted arc in a circle of radius 5 centimeters. Round to the nearest tenth.  Step 1: Convert degrees to radians.  Use to find the arc length.

14. Objective: 6-1 Angles and Radian Measure 14 You Try  Given a central angle of 125 o, find the length of its intercepted arc in a circle of radius 7 centimeters. Round to the nearest tenth.

15. Objective: 6-1 Angles and Radian Measure 15 Application of Arc Length  Winnipeg, Manitoba, Canada, and Dallas, Texas, lie along the 97 o W longitude line. The latitude of Winnipeg is 50 o N, and the latitude of Dallas is 33 o N. The radius of Earth is about 3960 miles. Find the approximate distance between the two cities.

16. Objective: 6-1 Angles and Radian Measure 16 You Try  The Swiss have long been highly regarded as makers of fine watches. The central angle formed by the hands of the watch on “12” and “5” is 150 o. The radius of the minute hand is ¾ centimeter. Find the distance traversed by the end of the minute hand from “12” to “5” to the nearest hundredth of a centimeter.

17. Objective: 6-1 Angles and Radian Measure 17 Area of a Circular Sector  If is the measure of the central angle expressed in radians and r is the measure of the radius of the circle, then the area of the sector, A, is as follows:

18. Objective: 6-1 Angles and Radian Measure 18 An Example  Find the area of a sector if the central angle measures radians and the radius of the circle is 16 centimeters. Round to the nearest tenth.

19. Objective: 6-1 Angles and Radian Measure 19 You Try  Find the area of a sector if the central angle measures radians and the radius of the circle is 11 centimeters. Round to the nearest tenth.

20. Objective: 6-1 Angles and Radian Measure 20 Homework  Homework 1: page 348, 16-36 even, 40, 42.  Homework 2: page 349, 44-52 even, 55, 57 and challenge problem 54.