# Solutions to Exercises, Section 5.2

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Instructor s Solutions Manual, Section 5.2 Exercise 1 Solutions to Exercises, Section 5.2 In Exercises 1 8, convert each angle to radians Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 15, obtaining 15 = 15π 180 radians = π 12 radians.

2 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 40, obtaining 40 = 40π 180 radians = 2π 9 radians.

3 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 45, obtaining 45 = 45π 180 radians = π 4 radians.

4 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 60, obtaining 60 = 60π 180 radians = π 3 radians.

5 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 270, obtaining 270 = 270π 180 radians = 3π 2 radians.

6 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 240, obtaining 240 = 240π 180 radians = 4π 3 radians.

7 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 1080, obtaining 1080 = 1080π 180 radians = 6π radians.

8 Instructor s Solutions Manual, Section 5.2 Exercise Start with the equation Divide both sides by 360 to obtain 360 = 2π radians. 1 = π 180 radians. Now multiply both sides by 1440, obtaining 1440 = 1440π 180 radians = 8π radians.

9 Instructor s Solutions Manual, Section 5.2 Exercise 9 In Exercises 9 16, convert each angle to degrees. 9. 4π radians Start with the equation Multiply both sides by 2, obtaining 2π radians = π radians = = 720.

10 Instructor s Solutions Manual, Section 5.2 Exercise π radians Start with the equation Multiply both sides by 3, obtaining 2π radians = π radians = = 1080.

11 Instructor s Solutions Manual, Section 5.2 Exercise π 9 radians Start with the equation 2π radians = 360. Divide both sides by 2 to obtain π radians = 180. Now divide both sides by 9, obtaining π radians = = 20. 9

12 Instructor s Solutions Manual, Section 5.2 Exercise π 10 radians Start with the equation 2π radians = 360. Divide both sides by 2 to obtain π radians = 180. Now divide both sides by 10, obtaining π radians = =

13 Instructor s Solutions Manual, Section 5.2 Exercise radians Start with the equation Divide both sides by 2π to obtain 2π radians = radian = 180. π Now multiply both sides by 3, obtaining 3 radians = π = 540. π

14 Instructor s Solutions Manual, Section 5.2 Exercise radians Start with the equation Divide both sides by 2π to obtain 2π radians = radian = 180. π Now multiply both sides by 5, obtaining 5 radians = π = 900. π

15 Instructor s Solutions Manual, Section 5.2 Exercise π 3 radians Start with the equation 2π radians = 360. Divide both sides by 2 to obtain π radians = 180. Now multiply both sides by 2 3, obtaining 2π 3 radians = = 120.

16 Instructor s Solutions Manual, Section 5.2 Exercise π 4 radians Start with the equation 2π radians = 360. Divide both sides by 2 to obtain π radians = 180. Now multiply both sides by 3 4, obtaining 3π 4 radians = = 135.

17 Instructor s Solutions Manual, Section 5.2 Exercise Suppose an ant walks counterclockwise on the unit circle from the point (0, 1) to the endpoint of the radius that forms an angle of 5π 4 radians with the positive horizontal axis. How far has the ant walked? The radius whose endpoint equals (0, 1) makes an angle of π 2 radians with the positive horizontal axis. This radius corresponds to the smaller angle shown below. Because 5π 4 = π + π 4, the radius that forms an angle of 5π 4 radians with the positive horizontal axis lies π 4 radians beyond the negative horizontal axis (half-way between the negative horizontal axis and the negative vertical axis). Thus the ant ends its walk at the endpoint of the radius corresponding to the larger angle shown below: 1 The ant walks along the thickened circular arc shown above. This circular arc corresponds to an angle of 5π 4 π 2 radians, which equals 3π 4 radians. Thus the distance walked by the ant is 3π 4.

18 Instructor s Solutions Manual, Section 5.2 Exercise Suppose an ant walks counterclockwise on the unit circle from the point ( 1, 0) to the endpoint of the radius that forms an angle of 6 radians with the positive horizontal axis. How far has the ant walked? The radius whose endpoint equals ( 1, 0) makes an angle of π radians with the positive horizontal axis. This radius lies on the negative horizontal axis. Because 2π 6.28, the radius that forms an angle of 6 radians with the positive horizontal axis lies slightly below the positive horizontal axis, as shown below: 1 The ant walks along the thickened circular arc shown above. This circular arc corresponds to an angle of 6 π radians. Thus the distance walked by the ant is 6 π.

19 Instructor s Solutions Manual, Section 5.2 Exercise Find the lengths of both circular arcs of the unit circle connecting the point (1, 0) and the endpoint of the radius that makes an angle of 3 radians with the positive horizontal axis. Because 3 is a bit less than π, the radius that makes an angle of 3 radians with the positive horizontal axis lies a bit above the negative horizontal axis, as shown below. The thickened circular arc corresponds to an angle of 3 radians and thus has length 3. The entire unit circle has length 2π. Thus the length of the other circular arc is 2π 3, which is approximately

20 Instructor s Solutions Manual, Section 5.2 Exercise Find the lengths of both circular arcs of the unit circle connecting the point (1, 0) and the endpoint of the radius that makes an angle of 4 radians with the positive horizontal axis. The radius that makes an angle of 4 radians with the positive horizontal axis is shown below. The thickened circular arc corresponds to an angle of 4 radians and thus has length 4. The entire unit circle has length 2π. Thus the length of the other circular arc is 2π 4, which is approximately

21 Instructor s Solutions Manual, Section 5.2 Exercise Find the lengths of both circular arcs of the unit circle connecting the point ( 2 2, 2 ) 2 and the point whose radius makes an angle of 1 radian with the positive horizontal axis. ( 2 2, 2 2 The radius of the unit circle whose endpoint equals ) π makes an angle of 4 radians with the positive horizontal axis, as shown with the clockwise arrow below. The radius that makes an angle of 1 radian with the positive horizontal axis is shown with a counterclockwise arrow. 1 Thus the thickened circular arc above corresponds to an angle of 1 + π 4 and thus has length 1 + π 4, which is approximately The entire unit circle has length 2π. Thus the length of the other circular arc below is 2π (1 + π 4 ), which equals 7π 4 1, which is approximately 4.50.

22 Instructor s Solutions Manual, Section 5.2 Exercise Find the lengths of both circular arcs of the unit circle connecting the point ( 2 2, 2 ) 2 and the point whose radius makes an angle of 2 radians with the positive horizontal axis. ( 2 2, 2 2 The radius of the unit circle whose endpoint equals ) 3π makes an angle of 4 radians with the positive horizontal axis, as shown with the longer arrow below. The radius that makes an angle of 2 radian with the positive horizontal axis is shown with the shorter arrow below. 1 Thus the thickened circular arc above corresponds to an angle of 3π 4 2 and thus has length 3π 4 2, which is approximately The entire unit circle has length 2π. Thus the length of the other circular arc below is 2π ( 3π 4 2), which equals 5π 4 + 2, which is approximately

23 Instructor s Solutions Manual, Section 5.2 Exercise For a 16-inch pizza, find the area of a slice with angle 3 4 radians. Pizzas are measured by their diameters; thus this pizza has a radius of 8 inches. Thus the area of the slice is , which equals 24 square inches.

24 Instructor s Solutions Manual, Section 5.2 Exercise For a 14-inch pizza, find the area of a slice with angle 4 5 radians. Pizzas are measured by their diameters; thus this pizza has a radius of 7 inches. Thus the area of the slice is , which equals square inches. 98 5

25 Instructor s Solutions Manual, Section 5.2 Exercise Suppose a slice of a 12-inch pizza has an area of 20 square inches. What is the angle of this slice? This pizza has a radius of 6 inches. Let θ denote the angle of this slice, measured in radians. Then 20 = 1 2 θ 62. Solving this equation for θ, wegetθ = 10 9 radians.

26 Instructor s Solutions Manual, Section 5.2 Exercise Suppose a slice of a 10-inch pizza has an area of 15 square inches. What is the angle of this slice? This pizza has a radius of 5 inches. Let θ denote the angle of this slice, measured in radians. Then 15 = 1 2 θ 52. Solving this equation for θ, wegetθ = 6 5 radians.

27 Instructor s Solutions Manual, Section 5.2 Exercise Suppose a slice of pizza with an angle of 5 6 radians has an area of 21 square inches. What is the diameter of this pizza? Let r denote the radius of this pizza. Thus 21 = r 2. Solving this equation for r, we get r = the pizza is approximately 14.2 inches Thus the diameter of

28 Instructor s Solutions Manual, Section 5.2 Exercise Suppose a slice of pizza with an angle of 1.1 radians has an area of 25 square inches. What is the diameter of this pizza? Let r denote the radius of this pizza. Thus 25 = r Solving this equation for r,wegetr = Thus the diameter of the pizza is approximately inches.

29 Instructor s Solutions Manual, Section 5.2 Exercise 29 For each of the angles in Exercises 29 34, find the endpoint of the radius of the unit circle that makes the given angle with the positive horizontal axis π 6 radians For this exercise it may be easier to convert to degrees. Thus we translate 5π 6 radians to 150. The radius making a 150 angle with the positive horizontal axis is shown below. The angle from this radius to the negative horizontal axis equals , which equals 30 as shown in the figure below. Drop a perpendicular line segment from the endpoint of the radius to the horizontal axis, forming a right triangle as shown below. We already know that one angle of this right triangle is 30 ; thus the other angle must be 60, as labeled below. The side of the right triangle opposite the 30 angle has length 1 2 ; the side of the right triangle opposite the 60 angle has length 3 2. Looking at the figure below, we see that the first coordinate of the endpoint of the radius is the negative of the length of the side opposite the 60 angle, and the second coordinate of the endpoint of the radius is the length of the side opposite the 30 angle. Thus the endpoint of the radius is ( 3 2, 1 ) 2.

30 Instructor s Solutions Manual, Section 5.2 Exercise

31 Instructor s Solutions Manual, Section 5.2 Exercise π 6 radians For this exercise it may be easier to convert to degrees. Thus we translate 7π 6 radians to 210. The radius making a 210 angle with the positive horizontal axis is shown below. The angle from the negative horizontal axis to this radius equals , which equals 30 as shown in the figure below. Draw a perpendicular line segment from the endpoint of the radius to the horizontal axis, forming a right triangle as shown below. We already know that one angle of this right triangle is 30 ; thus the other angle must be 60, as labeled below. The side of the right triangle opposite the 30 angle has length 1 2 ; the side of the right triangle opposite the 60 angle has length 3 2. Looking at the figure below, we see that the first coordinate of the endpoint of the radius is the negative of the length of the side opposite the 60 angle, and the second coordinate of the endpoint of the radius is the negative of the length of the side opposite the 30 angle. Thus the endpoint of the radius is ( 3 2, 1 ) 2.

32 Instructor s Solutions Manual, Section 5.2 Exercise

33 Instructor s Solutions Manual, Section 5.2 Exercise π 4 radians For this exercise it may be easier to convert to degrees. Thus we translate π 4 radians to 45. The radius making a 45 angle with the positive horizontal axis is shown below. Draw a perpendicular line segment from the endpoint of the radius to the horizontal axis, forming a right triangle as shown below. We already know that one angle of this right triangle is 45 ; thus the other angle must also be 45, as labeled below. The hypotenuse of this right triangle is a radius of the unit circle and thus has length 1. The other two sides each have length 2 2. Looking at the figure below, we see that the first coordinate of the endpoint of the 2 radius is 2 and the second coordinate of the endpoint of the radius is 2 2. Thus the endpoint of the radius is ( 2 2, 2 )

34 Instructor s Solutions Manual, Section 5.2 Exercise π 4 radians For this exercise it may be easier to convert to degrees. Thus we translate 3π 4 radians to 135. The radius making a 135 angle with the positive horizontal axis is shown below. Draw a perpendicular line segment from the endpoint of the radius to the horizontal axis, forming a right triangle as shown below. Because = 45, one angle of this triangle equals 45 ; thus the other angle must also be 45, as labeled below. The hypotenuse of this right triangle is a radius of the unit circle and thus has length 1. The other two sides each have length 2 2. Looking at the figure below, we see that the first coordinate of the endpoint of the 2 radius is 2 and the second coordinate of the endpoint of the radius 2 is also 2. Thus the endpoint of the radius is ( 2 2, 2 )

35 Instructor s Solutions Manual, Section 5.2 Exercise π 2 radians Note that 5π 2 = 2π + π 2. Thus the radius making an angle of 5π 2 radians with the positive horizontal axis is obtained by starting at the horizontal axis, making one complete counterclockwise rotation (which is 2π radians), and then continuing for another π 2 radians. The resulting radius is shown below. Its endpoint is (0, 1). 1

36 Instructor s Solutions Manual, Section 5.2 Exercise π 2 radians Note that 11π 2 = 4π + 3π 2. Thus the radius making an angle of 11π 2 radians with the positive horizontal axis is obtained by starting at the horizontal axis, making two complete counterclockwise rotations (which is 4π radians), and then continuing for another 3π 2 radians. The resulting radius is shown below. Its endpoint is (0, 1). 1

37 Instructor s Solutions Manual, Section 5.2 Problem 35 Solutions to Problems, Section 5.2 For each of the angles listed in Problems 35 42, sketch the unit circle and the radius that makes the indicated angle with the positive horizontal axis. Be sure to include an arrow to show the direction in which the angle is measured from the positive horizontal axis π 18 radians 5Π 18 1 An angle of 5π 18 radians, which equals 50.

38 Instructor s Solutions Manual, Section 5.2 Problem radian An angle of 1 2 radian, which approximately equals 28.6.

39 Instructor s Solutions Manual, Section 5.2 Problem radians 2 1 An angle of 2 radians, which approximately equals

40 Instructor s Solutions Manual, Section 5.2 Problem radians 5 1 An angle of 5 radians, which approximately equals

41 Instructor s Solutions Manual, Section 5.2 Problem π 5 radians 11Π 5 1 An angle of 11π 5 radians, which equals 396.

42 Instructor s Solutions Manual, Section 5.2 Problem π 12 radians Π 12 1 An angle of π 12 radians, which equals 15.

43 Instructor s Solutions Manual, Section 5.2 Problem radian 1 1 An angle of 1 radian, which approximately equals 57.3.

44 Instructor s Solutions Manual, Section 5.2 Problem π 9 radians 1 8Π 9 An angle of 8π 9 radians, which equals 160.

45 Instructor s Solutions Manual, Section 5.2 Problem Find the formula for the length of a circular arc corresponding to an angle of θ radians on a circle of radius r. Suppose 0 <θ 2π and consider a circular arc on a unit circle of radius r corresponding to an angle of θ radians, as shown in the thickened part of the unit circle below: Θ r The length of this circular arc equals the fraction of the entire circle taken up by this circular arc times the circumference of the entire circle. In other words, the length of this circular arc equals θ 2π times 2πr, which equals θr.

46 Instructor s Solutions Manual, Section 5.2 Problem Most dictionaries define acute angles and obtuse angles in terms of degrees. Restate these definitions in terms of radians. An angle between 0 radians and π 2 radians is called an acute angle. An angle between π 2 radians and π radians is called an obtuse angle.

47 Instructor s Solutions Manual, Section 5.2 Problem Find a formula (in terms of θ) for the area of the region bounded by the thickened radii and the thickened circular arc shown below. Θ 1 Here 0 <θ<2π and the radius shown above makes an angle of θ radians with the positive horizontal axis. The area of this region equals the fraction of the area inside the circle taken up by this region times the area inside the entire circle. In other words, the area of this region equals θ times π, which equals θ 2. 2π

48 Instructor s Solutions Manual, Section 5.2 Problem Suppose the region bounded by the thickened radii and circular arc shown above is removed. Find a formula (in terms of θ) for the perimeter of the remaining region inside the unit circle. The length of the thickened arc above is θ, and thus the length of the remaining part of the circle equals 2π θ. Each of the two radii has length 1. Thus the perimeter of the region in question is 2 + 2π θ.

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### 4.1: Angles and Radian Measure

4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is

### Solutions to Exercises, Section 6.1

Instructor s Solutions Manual, Section 6.1 Exercise 1 Solutions to Exercises, Section 6.1 1. Find the area of a triangle that has sides of length 3 and 4, with an angle of 37 between those sides. 3 4 sin

### 6.1: Angle Measure in degrees

6.1: Angle Measure in degrees How to measure angles Numbers on protractor = angle measure in degrees 1 full rotation = 360 degrees = 360 half rotation = quarter rotation = 1/8 rotation = 1 = Right angle

### Chapter 4. Trigonometric Functions. 4.1 Angle and Radian Measure /45

Chapter 4 Trigonometric Functions 4.1 Angle and Radian Measure 1 /45 Chapter 4 Homework 4.1 p290 13, 17, 21, 35, 39, 43, 49, 53, 57, 65, 73, 75, 79, 83, 89, 91 2 /45 Chapter 4 Objectives Recognize and

### Solutions Section J: Perimeter and Area

Solutions Section J: Perimeter and Area 1. The 6 by 10 rectangle below has semi-circles attached on each end. 6 10 a) Find the perimeter of (the distance around) the figure above. b) Find the area enclosed

### Remember that the information below is always provided on the formula sheet at the start of your exam paper

Maths GCSE Linear HIGHER Things to Remember Remember that the information below is always provided on the formula sheet at the start of your exam paper In addition to these formulae, you also need to learn

### 6.1 - Introduction to Periodic Functions

6.1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that

### MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 17 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Angles are what trigonometry is all about. This is where it all started, way back when.

Chapter 1 Tackling Technical Trig In This Chapter Acquainting yourself with angles Identifying angles in triangles Taking apart circles Angles are what trigonometry is all about. This is where it all started,

### Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

### A = ½ x b x h or ½bh or bh. Formula Key A 2 + B 2 = C 2. Pythagorean Theorem. Perimeter. b or (b 1 / b 2 for a trapezoid) height

Formula Key b 1 base height rea b or (b 1 / b for a trapezoid) h b Perimeter diagonal P d (d 1 / d for a kite) d 1 d Perpendicular two lines form a angle. Perimeter P = total of all sides (side + side

### Geometry SOL G.11 G.12 Circles Study Guide

Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter-10-circles-flashcardsflash-cards/).

### CIRCLES AND VOLUME Lesson 4: Finding Arc Lengths and Areas of Sectors

CIRCLES AND VOLUME Lesson : Finding Arc Lengths and Areas of Sectors Common Core Georgia Performance Standard MCC9 12.G.C.5 Essential Questions 1. How is radian measure related to degree measure? 2. How

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### Objective: To distinguish between degree and radian measure, and to solve problems using both.

CHAPTER 3 LESSON 1 Teacher s Guide Radian Measure AW 3.2 MP 4.1 Objective: To distinguish between degree and radian measure, and to solve problems using both. Prerequisites Define the following concepts.

### Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

### Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain

### AREA AND PERIMETER OF COMPLEX PLANE FIGURES

AREA AND PERIMETER OF OMPLEX PLANE FIGURES AREA AND PERIMETER OF POLYGONAL FIGURES DISSETION PRINIPLE: Every polygon can be dissected (or broken up) into triangles (or rectangles), which have no interior

### FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.

Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is

### y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

### ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE

ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE Keywords: Central angle An angle whose vertex is the center of a circle and whose sides contain the radii of the circle Arc Two points on a circle

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

### Upper Elementary Geometry

Upper Elementary Geometry Geometry Task Cards Answer Key The unlicensed photocopying, reproduction, display, or projection of the material, contained or accompanying this publication, is expressly prohibited

### Perimeter and area formulas for common geometric figures:

Lesson 10.1 10.: Perimeter and Area of Common Geometric Figures Focused Learning Target: I will be able to Solve problems involving perimeter and area of common geometric figures. Compute areas of rectangles,

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 3. Form: 201

Student Name: Teacher: District: Date: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 3 Description: Geometry Topic 6: Circles Form: 201 1. Which method is valid for proving

### DETAILED SOLUTIONS AND CONCEPTS - INTRODUCTION TO ANGLES

DETAILED SOLUTIONS AND CONCEPTS - INTRODUCTION TO ANGLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### *1. Understand the concept of a constant number like pi. Know the formula for the circumference and area of a circle.

Students: 1. Students deepen their understanding of measurement of plane and solid shapes and use this understanding to solve problems. *1. Understand the concept of a constant number like pi. Know the

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Chapter 6 Trigonometric Functions of Angles

6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

### vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector

vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector Convert Between DMS and Decimal Degree Form A. Write 329.125 in DMS form. First, convert 0.125

### Unit 1 - Radian and Degree Measure Classwork

Unit - Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side - starting and ending Position of the ray Standard position origin if the vertex,

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### Sect 10.1 Angles and Triangles

75 Sect 0. Angles and Triangles Objective : Converting Angle Units Each degree can be divided into 60 minutes, denoted 60', and each minute can be divided into 60 seconds, denoted 60". Hence, = 60' and

### Geometry Concepts. Figures that lie in a plane are called plane figures. These are all plane figures. Triangle 3

Geometry Concepts Figures that lie in a plane are called plane figures. These are all plane figures. Polygon No. of Sides Drawing Triangle 3 A polygon is a plane closed figure determined by three or more

### 3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?

1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### Arc Length and Areas of Sectors

Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.

### Combining Algebra and Geometry. chapter

chapter René Descartes, who invented the coordinate system we use to graph equations, explaining his work to Queen Christina of Sweden (from an 8 th - century painting by Dumesnil). Combining Algebra and

### Line AB (no Endpoints) Ray with Endpoint A. Line Segments with Endpoints A and B. Angle is formed by TWO Rays with a common Endpoint.

Section 8 1 Lines and Angles Point is a specific location in space.. Line is a straight path (infinite number of points). Line Segment is part of a line between TWO points. Ray is part of the line that

### September 14, Conics. Parabolas (2).notebook

9/9/16 Aim: What is parabola? Do now: 1. How do we define the distance from a point to the line? Conic Sections are created by intersecting a set of double cones with a plane. A 2. The distance from the

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### θ. The angle is denoted in two ways: angle θ

1.1 Angles, Degrees and Special Triangles (1 of 24) 1.1 Angles, Degrees and Special Triangles Definitions An angle is formed by two rays with the same end point. The common endpoint is called the vertex

### Axis of a coordination grid either of the two numgber lines used to form a coordinate grid. Plural is axes.

Axis of a coordination grid either of the two numgber lines used to form a coordinate grid. Plural is axes. Base Line a line or number used as a base for measurement or comparison An average man, 6 feet

### Linear Speed and Angular Speed

Preliminaries and Objectives Preliminaries: Circumference of a circle Conversion factors (dimensional analysis) Objectives: Given the central angle and radius (or diameter) of a circle, find the arc length

### THE DISTANCE FORMULA

THE DISTANCE FORMULA In this activity, you will develop a formula for calculating the distance between any two points in a coordinate plane. Part 1: Distance Along a Horizontal or Vertical Line To find

### 10.1: Areas of Parallelograms and Triangles

10.1: Areas of Parallelograms and Triangles Important Vocabulary: By the end of this lesson, you should be able to define these terms: Base of a Parallelogram, Altitude of a Parallelogram, Height of a

### Working in 2 & 3 dimensions Revision Guide

Tips for Revising Working in 2 & 3 dimensions Make sure you know what you will be tested on. The main topics are listed below. The examples show you what to do. List the topics and plan a revision timetable.

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### Name Class Date 1 E. Finding the Ratio of Arc Length to Radius

Name Class Date 0- Right-Angle Trigonometry Connection: Radian Measure Essential question: What is radian, and how are radians related to degrees? CC.9 2.G.C.5 EXPLORE Finding the Ratio of Arc Length to

### Trigonometry Basics. Angle. vertex at the and initial side along the. Standard position. Positive and Negative Angles

Name: Date: Pd: Trigonometry Basics Angle terminal side direction of rotation arrow CCW is positive CW is negative vertex initial side Standard position vertex at the and initial side along the Positive

### Unit 7 Syllabus: Area

Date Period Day Topic Unit 7 Syllabus: Area 1 Areas of Parallelograms and Triangles 2 Areas of Trapezoids, Rhombuses and Kites 3 Areas of Regular Polygons 4 Quiz 5 Perimeters and Areas of Similar Figures

### CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

### Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.

Chapter 11: Areas of Plane Figures (page 422) 11-1: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length

### Set 1: Circumference, Angles, Arcs, Chords, and Inscribed Angles

Goal: To provide opportunities for students to develop concepts and skills related to circumference, arc length, central angles, chords, and inscribed angles Common Core Standards Congruence Experiment

### LEVEL G, SKILL 1. Answers Be sure to show all work.. Leave answers in terms of ϖ where applicable.

Name LEVEL G, SKILL 1 Class Be sure to show all work.. Leave answers in terms of ϖ where applicable. 1. What is the area of a triangle with a base of 4 cm and a height of 6 cm? 2. What is the sum of the

### 5.1-Angles and Their Measure

5.1-Angles and Their Measure Objectives: 1. Find the degree or radian measure of co-terminal angles. 2. Convert between degrees minutes and seconds and decimal degrees. 3. Convert between degrees and radians.

### FS Geometry EOC Review

MAFS.912.G-C.1.1 Dilation of a Line: Center on the Line In the figure, points A, B, and C are collinear. http://www.cpalms.org/public/previewresource/preview/72776 1. Graph the images of points A, B, and

### CONNECT: Volume, Surface Area

CONNECT: Volume, Surface Area 2. SURFACE AREAS OF SOLIDS If you need to know more about plane shapes, areas, perimeters, solids or volumes of solids, please refer to CONNECT: Areas, Perimeters 1. AREAS

### Name: Class: Date: Geometry Chapter 3 Review

Name: Class: Date: ID: A Geometry Chapter 3 Review. 1. The area of a rectangular field is 6800 square meters. If the width of the field is 80 meters, what is the perimeter of the field? Draw a diagram

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### Mid-Chapter Quiz: Lessons 4-1 through 4-4

Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to

### Trigonometric Functions: The Unit Circle

Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry

### 8-3 Perimeter and Circumference

Learn to find the perimeter of a polygon and the circumference of a circle. 8-3 Perimeter Insert Lesson and Title Circumference Here perimeter circumference Vocabulary The distance around a geometric figure

### Lecture Two Trigonometric

Lecture Two Trigonometric Section.1 Degrees, Radians, Angles and Triangles Basic Terminology Two distinct points determine line AB. Line segment AB: portion of the line between A and B. Ray AB: portion

### Taxicab Geometry, or When a Circle is a Square. Circle on the Road 2012, Math Festival

Taxicab Geometry, or When a Circle is a Square Circle on the Road 2012, Math Festival Olga Radko (Los Angeles Math Circle, UCLA) April 14th, 2012 Abstract The distance between two points is the length

### Name Geometry Exam Review #1: Constructions and Vocab

Name Geometry Exam Review #1: Constructions and Vocab Copy an angle: 1. Place your compass on A, make any arc. Label the intersections of the arc and the sides of the angle B and C. 2. Compass on A, make

### Which statement best describes the shaded turn?

Name: Class: Date: ID: A Chapter 11 Practice Test 1. Use the diagram to answer the question. Which statement best describes the shaded turn? A. B. C. D. 1 turn clockwise 4 1 turn counterclockwise 4 3 turn

### Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.

Pre-Calculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle

### Maths Class 11 Chapter 3. Trigonometric Functions

1 P a g e Angle Maths Class 11 Chapter 3. Trigonometric Functions When a ray OA starting from its initial position OA rotates about its end point 0 and takes the final position OB, we say that angle AOB

### Spherical coordinates 1

Spherical coordinates 1 Spherical coordinates Both the earth s surface and the celestial sphere have long been modeled as perfect spheres. In fact, neither is really a sphere! The earth is close to spherical,

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

### Circular Motion. Physics 1425 Lecture 18. Michael Fowler, UVa

Circular Motion Physics 1425 Lecture 18 Michael Fowler, UVa How Far is it Around a Circle? A regular hexagon (6 sides) can be made by putting together 6 equilateral triangles (all sides equal). The radius

### Rugs. This problem gives you the chance to: find perimeters of shapes use Pythagoras Rule. Hank works at a factory that makes rugs.

Rugs This problem gives you the chance to: find perimeters of shapes use Pythagoras Rule Hank works at a factory that makes rugs. The edge of each rug is bound with braid. Hank s job is to cut the correct

5.1 The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives The Unit Circle Terminal Points on the Unit Circle The Reference Number 2 The Unit Circle In this section we explore some

### Covering & Surrounding Notes Problem 1.1

Problem 1.1 Definitions: Area - the measure of the amount of surface enclosed by the sides of a figure. Perimeter - the measure of the distance around a figure. Bumper-cars is one of the most popular rides

### 17.1 Cross Sections and Solids of Rotation

Name Class Date 17.1 Cross Sections and Solids of Rotation Essential Question: What tools can you use to visualize solid figures accurately? Explore G.10.A Identify the shapes of two-dimensional cross-sections

Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

### Geometry B Solutions. Written by Ante Qu

Geometry B Solutions Written by Ante Qu 1. [3] During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram.

### Pythagorean theorems in the alpha plane

MATHEMATICAL COMMUNICATIONS 211 Math. Commun., Vol. 14, No. 2, pp. 211-221 (2009) Pythagorean theorems in the alpha plane Harun Baris Çolakoğlu 1,, Özcan Gelişgen1 and Rüstem Kaya 1 1 Department of Mathematics

### 5.1 Angles and Their Measure. Objectives

Objectives 1. Convert between decimal degrees and degrees, minutes, seconds measures of angles. 2. Find the length of an arc of a circle. 3. Convert from degrees to radians and from radians to degrees.

### Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings

Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings Trigonometry in Right Triangles A In right ABC, AC is called the hypotenuse. The vertices are labelled using capital letters.

### CCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name:

GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M9-1.G..1 Prove that all circles are similar. M9-1.G.. Identify and describe relationships

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, :15 a.m. SAMPLE RESPONSE SET

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Question 29................... 2 Question 30...................

### Math 366 Lecture Notes Section 11.1 Basic Notions (of Geometry)

Math 366 Lecture Notes Section. Basic Notions (of Geometry) The fundamental building blocks of geometry are points, lines, and planes. These terms are not formally defined, but are described intuitively.

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### 1. AREA BETWEEN the CURVES

1 The area between two curves The Volume of the Solid of revolution (by slicing) 1. AREA BETWEEN the CURVES da = {( outer function ) ( inner )} dx function b b A = da = [y 1 (x) y (x)]dx a a d d A = da

### 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft

2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### are radii of the same circle. So, they are equal in length. Therefore, DN = 8 cm.

For Exercises 1 4, refer to. 1. Name the circle. The center of the circle is N. So, the circle is 2. Identify each. a. a chord b. a diameter c. a radius a. A chord is a segment with endpoints on the circle.