5 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley.

Slides:



Advertisements
Similar presentations
Copyright © 2009 Pearson Education, Inc. CHAPTER 6: The Trigonometric Functions 6.1The Trigonometric Functions of Acute Angles 6.2Applications of Right.
Advertisements

5.3 Trigonometric Functions of Any Angle Tues Oct 28 Do Now Find the 6 trigonometric values for 60 degrees.
Lesson 5-1: Angles and Degree Measure
Radian and Degree Measure
Drill Calculate:.
Radian and Degree Measure Objectives: Describe Angles Use Radian and Degree measures.
SECTION 14-1 Angles and Their Measures Slide
Slide 1-1 By Y. Ath. Slide 1-2 Section 1 Angles Slide 1-3 Basic Terminology Line AB. Line segment AB Ray AB.
Copyright © Cengage Learning. All rights reserved. Trigonometric Functions: Right Triangle Approach.
Copyright © 2005 Pearson Education, Inc.. Chapter 1 Trigonometric Functions.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 Trigonometry with Ms. Miller Please find your name and the group number that it corresponds to on.
1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved.
Rev.S08 MAC 1114 Module 1 Trigonometric Functions.
1 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley.
Objectives Change from radian to degree measure, and vice versa Find angles that are co-terminal with a given angle Find the reference angle for a given.
Quick Crisp Review Radian Coterminal Angles Complementary and Supplementary.
Radian and Degree Measure. Radian Measure A radian is the measure of a central angle that intercepts an arc length equal to the radius of the circle Radians.
An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.
How do we draw angles in standard position?
1 Copyright © Cengage Learning. All rights reserved. 5. The Trigonometric Functions 6.1 Angles and Their Measures.
Section 6.3 Trigonometric Functions of Any Angle Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Copyright © 2005 Pearson Education, Inc.. Introduction to Trigonometry Angle Relationships and Similar Triangles.
An angle is formed by rotating an initial arm about a fixed point. Angles in Standard Position - Definitions An angle is said to be in standard position.
Copyright © 2007 Pearson Education, Inc. Slide Angles and Arcs Basic Terminology –Two distinct points A and B determine the line AB. –The portion.
Chapter 4 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Angles and Radian Measure.
Copyright © Cengage Learning. All rights reserved. Trigonometric Functions: Right Triangle Approach.
Chapter 7: Trigonometric Functions Section 7.1: Measurement of Angles.
1 Copyright © Cengage Learning. All rights reserved. 1 Trigonometry.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
5.1 The Unit Circle.
Measurement of Rotation
Copyright © 2014 Pearson Education, Inc.
Degrees and Radians Pre-Calculus Keeper 11.
Radian and Degree Measure
Radian and Degree Measure
MATH 1330 Section 4.3.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Coterminal Angles.
Chapter 1 Trigonometric Functions.
Radian and Degree Measure
Angle Measure In this case, R1 is called the initial side, and R2 is called the terminal side of the angle. If the rotation is counterclockwise, the angle.
Trigonometric Functions of Any Angle
1.1 Angles Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved.
10-2 Angles of Rotation Warm Up Lesson Presentation Lesson Quiz
Radian Measure and Coterminal Angles
Radian and Degree Measures
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Students, Take out your calendar and your homework. Take out your spiral notebook and Complete the DNA. Use your notes if necessary. Simplify.
Evaluating Trigonometric Functions
Chapter 8: The Unit Circle and the Functions of Trigonometry
5.1 The Unit Circle.
Angles and Their Measures
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Angles and Their Measures
Section 4.3 Trigonometric Functions of Angles
Angles and Radian Measure
Precalculus Essentials
MATH 1330 Section 4.3.
Do Now Find the measure of the supplement for each given angle.
Welcome to Trigonometry!
Copyright © Cengage Learning. All rights reserved.
Chapter 5 – Trigonometric Functions
Chapter 8: The Unit Circle and the Functions of Trigonometry
Copyright © Cengage Learning. All rights reserved.
1 Trigonometric Functions.
13-2 Angles and Angle Measure
Do Now Classify the following angles as obtuse, right, straight, or acute
Presentation transcript:

5 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley

Trigonometric Functions 5 5.1 Angles 5.2 Trigonometric Functions 5.3 Evaluating Trigonometric Functions 5.4 Solving Right Triangles Copyright © 2009 Pearson Addison-Wesley

5.1 Angles Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles Copyright © 2009 Pearson Addison-Wesley 1.1-3

Basic Terminology Two distinct points determine line AB. Line segment AB—a portion of the line between A and B, including points A and B. Ray AB—portion of line AB that starts at A and continues through B, and on past B. Copyright © 2009 Pearson Addison-Wesley

Basic Terminology An angle consists of two rays in a plane with a common endpoint. The two rays are the sides of the angle. The common endpoint is called the vertex of the angle. Copyright © 2009 Pearson Addison-Wesley

Basic Terminology An angle’s measure is generated by a rotation about the vertex. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle. Copyright © 2009 Pearson Addison-Wesley

Basic Terminology Positive angle: The rotation of the terminal side of an angle is counterclockwise. Negative angle: The rotation of the terminal side is clockwise. Copyright © 2009 Pearson Addison-Wesley

Types of Angles The most common unit for measuring angles is the degree. A complete rotation of a ray gives an angle whose measure is 360°. of complete rotation gives an angle whose measure is 1°. Copyright © 2009 Pearson Addison-Wesley

Angles are classified by their measures. Types of Angles Angles are classified by their measures. Copyright © 2009 Pearson Addison-Wesley

Example 1 FINDING THE COMPLEMENT AND THE SUPPLEMENT OF AN ANGLE For an angle measuring 40°, find the measure of its (a) complement and (b) supplement. (a) To find the measure of its complement, subtract the measure of the angle from 90°. Complement of 40°. (b) To find the measure of its supplement, subtract the measure of the angle from 180°. Supplement of 40°. Copyright © 2009 Pearson Addison-Wesley 1.1-10

Find the measure of each marked angle. Example 2 FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES Find the measure of each marked angle. Since the two angles form a right angle, they are complementary. Combine terms. Divide by 9. Determine the measure of each angle by substituting 10 for x: Copyright © 2009 Pearson Addison-Wesley 1.1-11

Find the measure of each marked angle. Example 2 FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES (continued) Find the measure of each marked angle. Since the two angles form a straight angle, they are supplementary. The angle measures are and . Copyright © 2009 Pearson Addison-Wesley 1.1-12

Degrees, Minutes, Seconds One minute is 1/60 of a degree. One second is 1/60 of a minute. Copyright © 2009 Pearson Addison-Wesley

Perform each calculation. Example 3 CALCULATING WITH DEGREES, MINUTES, AND SECONDS Perform each calculation. (a) (b) Add degrees and minutes separately. Write 90° as 89°60′. Copyright © 2009 Pearson Addison-Wesley 1.1-14

(a) Convert 74°8′14″ to decimal degrees to the nearest thousandth. Example 4 CONVERTING BETWEEN DECIMAL DEGREES AND DEGREES, MINUTES, AND SECONDS (a) Convert 74°8′14″ to decimal degrees to the nearest thousandth. Copyright © 2009 Pearson Addison-Wesley 1.1-15

(b) Convert 34.817° to degrees, minutes, and seconds. Example 4 CONVERTING BETWEEN DECIMAL DEGREES AND DEGREES, MINUTES, AND SECONDS (continued) (b) Convert 34.817° to degrees, minutes, and seconds. Copyright © 2009 Pearson Addison-Wesley 1.1-16

Standard Position An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. Copyright © 2009 Pearson Addison-Wesley

Quandrantal Angles Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles. Copyright © 2009 Pearson Addison-Wesley 1.1-18

Coterminal Angles A complete rotation of a ray results in an angle measuring 360. By continuing the rotation, angles of measure larger than 360 can be produced. Such angles are called coterminal angles. The measures of coterminal angles differ by 360. Copyright © 2009 Pearson Addison-Wesley

An angle of 908° is coterminal with an angle of 188°. Example 5 FINDING MEASURES OF COTERMINAL ANGLES Find the angle of least possible positive measure coterminal with an angle of 908°. Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°. An angle of 908° is coterminal with an angle of 188°. Copyright © 2009 Pearson Addison-Wesley 1.1-20

An angle of –75 ° is coterminal with an angle of 285°. Example 5 FINDING MEASURES OF COTERMINAL ANGLES (continued) Find the angle of least possible positive measure coterminal with an angle of –75°. Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°. An angle of –75 ° is coterminal with an angle of 285°. Copyright © 2009 Pearson Addison-Wesley 1.1-21

The least integer multiple of 360° greater than 800° is Example 5 FINDING MEASURES OF COTERMINAL ANGLES (continued) Find the angle of least possible positive measure coterminal with an angle of –800°. The least integer multiple of 360° greater than 800° is An angle of –800° is coterminal with an angle of 280°. Copyright © 2009 Pearson Addison-Wesley 1.1-22

Coterminal Angles To find an expression that will generate all angle coterminal with a given angle, add integer multiples of 360° to the given angle. For example, the expression for all angles coterminal with 60° is Copyright © 2009 Pearson Addison-Wesley

Coterminal Angles Copyright © 2009 Pearson Addison-Wesley

Example 6 ANALYZING THE REVOLUTIONS OF A CD PLAYER CAV (Constant Angular Velocity) CD players always spin at the same speed. Suppose a CAV player makes 480 revolutions per minute. Through how many degrees will a point on the edge of a CD move in two seconds? Copyright © 2009 Pearson Addison-Wesley 1.1-25

In two seconds, the player will revolve times. Example 6 ANALYZING THE REVOLUTIONS OF A CD PLAYER Solution The player revolves 480 times in one minute or times = 8 times per second. In two seconds, the player will revolve times. Each revolution is 360°, so a point on the edge of the CD will revolve in two seconds. Copyright © 2009 Pearson Addison-Wesley 1.1-26

Homework pp. 499–502 (1–8, 13–15, 23–32 , 78, 79, 83, 87, 89–92, 105, 108, 109, 113, 130, 131) Copyright © 2009 Pearson Addison-Wesley 1.1-27