CHAPTER 4: Trigonometry and the Unit Circle Section 4.1: Angles and Angle Measure

Transcription

1 CHAPTER 4: Trigonometry and the Unit Circle Section 4.1: Angles and Angle Measure 1

2 (A) Standard Position When drawing an angle θ on the x y plane in standard position, the following conditions must apply: Vertex must be at The initial arm lies on Angles are often classified according to the quadrant in which their terminal sides lie. 2

3 (B) Positive and Negative Rotation (Standard Position) (C) Reference Angle the acute angle that is formed by the terminal arm of the angle and either the positive or negative x axis. 3

4 Ex)Sketch in standard position the following angles and identify the type of angle, its reference angle. A) 300 B) 200 C) 800 D) 500 Note: For angles larger than 360º or smaller than 360º we subtract or add multiples of 360º to determine where the angle is. 4

5 There are two units for measuring angles:» Degrees» Radians Angle Measure What is a degree? An angle measurement One degree is defined as of a full rotation. What is a radian? A radian is an angle measurement that gives the ratio: Each full radian measure occurs when the arc length is the same as the length of the radius 5

6 Angle in radian Explanation 6

7 When dealing with circles let's start with a special one, the unit circle.» The unit circle has a radius of 1 If the radius is 1 what is the angle of in radians? Common Radian Measure Angle Degrees Angle Radian 7

8 Ex) Convert the following to radian measure (exact and approximate). A) 135 B) 58 C) 225 D) 144 E) F) 118 8

9 9

10 Ex)Sketch the following angles in standard position. 10

11 Co terminal Angles angles in standard position with the same terminal arm and can be measured in degrees or radians Co terminal angles can be found by adding or subtracting multiples of 360 or 2π In general, if θ is an angle in standard position then any angle of the form: θ ± 2πn, where n N or θ ± 360 n, where n N 11

12 Ex)Find an angle that is co terminal with each of the following angles. Sketch to check your answer. A) 112 B) 700 Find one negative and one positive angle that is a co terminal with each angle. A) 515 B) C) D) 3 12

13 Ex) Determine the measures of all angles that are co terminal with 120 in the given domain. A) 0 θ 720 B) 360 θ 0 13

14 Ex)Write an expression for all of the angles co terminal with each angle. Indicate what your variable represents. A) 310 B) 14

15 Arc Length, Radius and the Radian Measure of the Central Angle 2 types of arc length: > Minor Arc > Major Arc Determine a formula relating the radius (r), central angle θ (measured in radians) and arc length of a circle (a). [Earlier in the notes, the arc length was denoted by the variable s] Arc Length = θ x radius a = θr Know how to rearrange!! Ex) Determine the measures of the arc length subtended by the angles and radii below: A) central angle of with radius of 10 cm. B) central angle of 2.6 rad with radius of 4.9cm. 15

16 Ex) Determine the measure of the radius of a circle in the following diagram. Ex) An arc of 20 cm in length cuts a circle of radius 5.4 cm. Determine the measure of the central angle in radians and degrees. 16

17 Ex) Find the radius of a circle in which an arc of 3 km subtends a central angle of 20 o. Ex) During a family vacation, you go to dinner at the Seattle Space Needle. There is a rotating restaurant at the top of the needle that is circular and has a radius of 40 feet. It makes one rotation per hour. At 6:42 p.m., you take a seat at a window table. You finish dinner at 8:28 p.m. Through what angle did your position rotate during your stay? How many feet did your position revolve? Do questions 12 and 13 page

18 18