Warm-up:

1. Warm-up: Evaluate the six trig functions at each real number. Try without notes.

2. Answers

6. PrecalculusLesson ( ) 4.1 Check: # 17-26,35-42

7. Objective (4.1) • Convert angle measures; Radians, Degrees, and Degrees, minutes, seconds. • Find coterminal, complementary, andsupplementary angles. Lesson Powerpoint & hand-out notes 4.1

8. Notes 4.1 Radian Measure In addition to the degree, there is a second unit of angle measure called the radian. Radian measure is the unit of angle measure used frequently in trigonometry, calculus, and more advanced mathematics. B is a central angle whose sides intersect the circle at points A and B. If the length of arc AB is equal to the radius r of the circle, then has a measure of 1 radian. O A Definition: One radian (rad.) is the measure of a central angle whose intercepted arc has a length equal to the circle’s radius.

9. A circle with radius r = 1 is called a unit circle. The radian measure of a central angle of a unit circle is equal to the length of the intercepted arc. r r r On a unit circle the circumference (one revolution) is . This means that half of a revolution or 180 degrees = radians. Conversion between Radians and Degrees

10. Converting from Degrees to Radians Change each of the following common angles from degrees to radians.

11. Converting from Radians to Degrees Change each of the following angles from radians to degrees: Do EX 1

12. Angles and Their Measure In geometry an angle was defined as the union of two rays with a common endpoint called the vertex of the angle. In trigonometry an angle is described as a ray that rotates about the vertex. The beginning ray, called the initial side of the angle, is rotated about its endpoint. The final position is called the terminal side of the angle. Terminal side Initial side vertex The angle can be formed by rotating or counterclockwise. clockwise

13. An angle is usually named with a single Greek letter, such as (alpha) (beta) (gamma) or (theta) or by an uppercase letter such as A, B, or C. An angle is called an angle in standard position when it is positioned on a rectangular coordinate system with its vertex at the origin and its initial side on the positive x-axis.

14. Angles are often measured in units called degrees. One complete counterclockwise rotation is defined to be 360 degrees, denoted . Angles can make more than 1 revolution and therefore have measures greater than . When rotating counterclockwise, the measure is positive. When rotating clockwise, the angle measure is negative.

15. Two angles are coterminal if they have the same initial side and the same terminal side. Coterminal angles can be found by adding or subtracting an integer multiple of . Examples of coterminal angles:

16. Sketch each of the following angles in standard position and find one negative and one positive angle coterminal with each: is coterminal with: is coterminal with: is coterminal with: Do EX 2

17. Finding Complements and Supplements Supplementary- 2 angles whose measures add up to 180 degrees. Complementary- 2 angles whose measures add up to 90 degrees. Negative angles do not have complements or supplements. a) Find the complement of a angle. b) Find the supplement of a angle. Do EX 3

18. Arc Length Given a circle of radius r, if x is the length of the intercepted arc of a central angle , then Example 1: Find the length x of the arc of a circle of radius 2 intercepted by a central angle measuring . Example 2: Determine the length of the arc of a circle with . Do EX 5

19. Do EX 4

20. Assignments Classwork p.288 (19-31 odd,43,59,63,81,91) Homework(4.1 ) P. 288 #18-33 (x3), 44,52,58,64,83,85,89,93,105

21. Closure