Unit 5 The Trigonometric Functions

Transcription

1 Unit 5 The Trigonometric Functions 21st Century Learning Expectations Academic Ellington High School Students: Create, perform or respond effectively. Read effectively. Write effectively. Reason effectively, think critically and solve problems. Utilize real-world digital and other technology effectively. Collaborate effectively. Civic and Social Ellington High School Students: Value and demonstrate personal responsibility, character, cultural understanding and ethical behavior. Jan 27 8:34 AM Trigonometry The simplest definition of trigonometry would go something like this: Trigonometry is the study of angles and triangles. The word trigonometry literally means "triangle measurement." However, the study of trigonometry is much more complex and comprehensive than that. In this unit alone, you will find trigonometric functions of angles, solve triangles, find the area of triangles and verify trigonometric identities. Trigonometry has applications in construction, geography, physics, acoustics, medicine, meteorology, and navigation, among other fields. Jan 27 11:04 AM 1

2 If I had a board that I wanted to stand up vertically.. and to make it stand on its own I was going to add a brace.. and I wanted the brace to be mounted at a 25 0 angle to the board...how long should the brace be? and where would it touch the floor? Jan 27 4:49 PM If it is low tide will the island be submerged at high tide? How high will the tide get? How far away would it be at low tide from the beach where these 2 guys are standing? Jan 27 4:54 PM 2

3 Section 5.1 Angles and Their Measure By the end of this section, you should know: The degree and radian measure of angles on a unit circle The difference between the initial and the terminal The standard position of an angle in the unit circle The location of a quadrantal angle The difference between positive and negative angles in a unit circle By the end of this section, you should be able to: Change from degrees, minutes and seconds to decimals Change from decimal degree measure to degrees, minutes, and seconds Change from radian to degree measure Change from degree to radian measure Find angles that are coterminal with a given angle Find the reference angle for a given angle Jan 27 8:36 AM Unit Circle Group Activity: Fill in the degree measure of the unit circle using a protractor and patterns. Jan 27 8:43 AM 3

4 What is a negative angle? Where would it be on the unit circle? Where would 380 o be located? Jan 29 4:54 AM Mar 24 7:19 AM 4

5 Decimals to degrees, minutes and seconds hp:// Example 1: Convert to degrees, minutes and seconds Jan 27 9:05 AM Example 2: Convert to degrees, minutes and seconds Jan 27 10:53 AM 5

6 Degrees, minutes and seconds to decimals hp:// Example 3: Convert ' 15.03" to decimal degrees Jan 27 9:04 AM Example 4: Convert ' 1.22" to decimal degrees Jan 27 10:53 AM 6

7 hp://itouchmap.com/latlong.html latude and longitude for EHS o 37 Maple St, Ellington, CT ' '' Jan 27 10:53 AM VOCAB Flashcards initial The fixed ray () of an angle. terminal The rotating ray () of an angle. positive anglethe angle formed if the rotation is in a counterclockwise direction. negative anglethe angle formed if the rotation is in a clockwise direction. standard position An angle with its vertex at the origin and its initial along the positive x axis quadrantal angle The type of angle formed if the terminal of an angle in standard position coincides with one of the axes. unit circlea circle of radius 1 whose center is at the origin of a rectangular coordinated system. reference anglethe acute angle formed by the terminal of a nonquadrantal angle in standard position and the x axis. coterminal angles Two angles in standard position and with the same terminal. Oct 5 11:01 AM 7

8 terminal initial initial terminal terminal initial Sep 13 2:08 PM Example 5: If each angle has the given measure and is in standard position, determine the quadrant in which its terminal lies. a. 210 b. 220 c. 15πd. 4π 4 3 Sep 13 2:13 PM 8

9 Coterminal angles α = greek letter alpha If α is the degree measure of an angle, then all angles of the form α + 360kº, where k is an integer, are coterminal with α. Oct 5 11:47 AM Example 6: Find one positive angle and one negative angle that are coterminal with an angle having measure 70 Oct 5 3:26 PM 9

10 Example 7: Identify all angles on the unit circle that are cotermina with a 60 angle. Oct 5 3:27 PM Reference Angles FORMAL RULE For any angle α, 0 < α < 360 o, its reference angle α' is defined by: a. α, when the terminal is in Quadrant I b. 180 o α, when the terminal is in Quadrant II c. α 180 o, when the terminal is in Quadrant III d. 360 o α, when the terminal is in Quadrant IV Oct 5 3:27 PM 10

11 Example 8: a. Find the reference angle for o75 b. Find the reference angle for o75 c. Find the reference angle for 135 o Oct 5 3:27 PM Wrap-up: What did you learn today? From what we learned in class today, what were you confident with? From what we learned in class today, what do you need help with? Homework 5.1 ID 1 & Degree Conversion Worksheets Jan 27 9:05 AM 11