# Mid-Chapter Quiz: Lessons 9-1 through 9-3

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Graph each point on a polar grid. 1. A( 2, 45 ) 3. Because = 45, locate the terminal side of a 45 angle with the polar axis as its initial side. Because r = 2, plot a point 2 units from the pole in the opposite direction of the terminal side of the angle. Because =, locate the terminal side of a - angle with the polar axis as its initial side. Because r = 1.5, plot a point 1.5 units from the pole in the opposite direction of the terminal side of the angle. 2. D(1, 315 ) Because = 315, locate the terminal side of a angle with the polar axis as its initial side. Because r = 1, plot a point 1 unit from the pole along the terminal side of the angle. esolutions Manual - Powered by Cognero Page 1

2 4. 6. Because =, locate the terminal side of a The solutions of = are ordered pairs of the - angle with the polar axis as its initial side. form, where r is any real number. The graph consists of all points on the line that make an angle of with the positive polar axis. Because r = 3, plot a point 3 units from the pole along the terminal side of the angle. 7. = 60 Graph each polar equation. 5. r = 3 The solutions of r = 3 are ordered pairs of the form (3, ), where is any real number. The graph consists of all points that are 3 units from the pole, so the graph is a circle centered at the origin with radius 3. The solutions of = 60 are ordered pairs of the form (r, 60 ), where r is any real number. The graph consists of all points on the line that make an angle of 60 with the positive polar axis. esolutions Manual - Powered by Cognero Page 2

4 Graph each equation. 10. r = sec Make a table of values to find the r-values corresponding to various values of on the interval [0, 2π]. Round each r-value to the nearest tenth. r = π π 0.3 sec 11. r = cos Because the polar equation is a function of the cosine function, it is symmetric with respect to the polar axis. Therefore, make a table and calculate the values of r on [0, π]. r = π 0.3 cos Use these points and polar axis symmetry to graph the function. Graph the ordered pairs (r, ) and connect them with a line. esolutions Manual - Powered by Cognero Page 4

5 12. r = 3 csc Make a table of values to find the r-values corresponding to various values of on the interval [0, 2π]. Round each r-value to the nearest tenth. r = 3 csc 0 π r = 4 sin Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = 4 sin Use these points and symmetry with respect to the line = to graph the function. Graph the ordered pairs (r, ) and connect them with a line. 14. STAINED GLASS A rose window is a circular window seen in gothic architecture. The pattern of the window radiates from the center. The window shown can be approximated by the equation r = 3 sin 6. Use symmetry, zeros, and maximum r-values of the function to graph the function. Refer to the image on Page 560. esolutions Manual - Powered by Cognero Page 5

6 Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Sketch the graph of the rectangular function y = 3 sin 6x on the interval. From the graph, you Use these and a few additional points to sketch the graph of the function. can see that = 3 when and y = 0 when Interpreting these results in terms of the polar equation r = 3 sin 6, we can say that has a maximum value of 3 when and r = 0 when Since the function is symmetric with respect to the line =, make a table and calculate the values of r on. r = 3 sin esolutions Manual - Powered by Cognero Page 6

7 Identify and graph each classic curve. 15. r = sin The equation is of the form r = a sin, so its graph is a circle. Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = sin r = + 3, 0 The equation is of the form r = a + b, so its graph is a spiral of Archimedes. Use points on the interval [0, 2π] to sketch the graph of the function. r = π π 5.1 Use these points and symmetry with respect to the line = to graph the function. esolutions Manual - Powered by Cognero Page 7

8 17. r = cos The equation is of the form r = a + b cos, so its graph is a limacon. Since a < b, the graph with have an inner loop. Because this polar equation is a function of the cosine function, it is symmetric with respect to the polar axis. Therefore, make a table and calculate the values of r on. r = cos 0 3 π Use these points and polar axis symmetry to graph the function. 18. r = 5 sin 3 The equation is of the form r = a sin n, so its graph is a rose. Because this polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = 5 sin Use these points and symmetry with respect to the line = to graph the function. esolutions Manual - Powered by Cognero Page 8

9 19. MULTIPLE CHOICE Identify the polar graph of y 2 = x. 20. Find the rectangular coordinates for each point with the given polar coordinates. For, r = 4 and =. Write the rectangular equation y 2 = x in polar The rectangular coordinates of. are form. 21. For, r = 2 and =. Graph r = cos csc 2 using a graphing calculator. Let = and solve for r. The rectangular coordinates of. are The point corresponds to graph B. The correct answer is B. esolutions Manual - Powered by Cognero Page 9

10 22. ( 1, 210 ) For ( 1, 210 ), r = 1 and = 210. Find two pairs of polar coordinates for each point with the given rectangular coordinates if 0 2π. Round to the nearest hundredth. 24. ( 3, 5) For ( 3, 5), x = 3 and y = 5. Since x < 0, use to find. The rectangular coordinates of ( 1, 210 ) are. 23. (3, 30 ) For (3, 30 ), r = 3 and = 30. One set of polar coordinates is (5.83, 2.11). Another representation that uses a negative r-value is ( 5.83, π) or ( 5.83, 5.25). 25. (8, 1) The rectangular coordinates of (3, 30 ) are. For (8, 1), x = 8 and y = 1. Since x > 0, use to find. One set of polar coordinates is (8.06, 0.12). Another representation that uses a negative r-value is ( 8.06, π) or ( 8.06, 3.27). esolutions Manual - Powered by Cognero Page 10

11 26. (7, 6) Write a rectangular equation for each graph. For (7, 6), x = 7 and y = 6. Since x > 0, use to find. 28. One set of polar coordinates is (9.22, 0.71). Since this set is not in the required domain, two more sets have to be found. A representation that uses a positive r-value is (9.22, π) or (9.22, 5.57). A representation that uses a negative r-value is ( 9.22, π) or ( 9.22, 2.43). 27. ( 4, 10) For ( 4, 10), x = 4 and y = 10. Since x < 0, use tan 1 + π to find. 29. One set of polar coordinates is (10.77, 4.33). Another representation that uses a negative r-value is ( 10.77, 4.33 π) or ( 10.77, 1.19). esolutions Manual - Powered by Cognero Page 11

### Complex Numbers, Polar Equations, and Parametric Equations. Copyright 2017, 2013, 2009 Pearson Education, Inc.

8 Complex Numbers, Polar Equations, and Parametric Equations Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 8.5 Polar Equations and Graphs Polar Coordinate System Graphs of Polar Equations Conversion

9.5 Polar Coordinates Copyright Cengage Learning. All rights reserved. Introduction Representation of graphs of equations as collections of points (x, y), where x and y represent the directed distances

### Pre-Calc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015

Pre-Calc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015 Date Objective/ Topic Assignment Did it Monday Polar Discovery Activity pp. 4-5 April 27 th Tuesday April 28 th Converting between

### Polar Coordinates. OpenStax. 1 Dening Polar Coordinates

OpenStax-CNX module: m53852 1 Polar Coordinates OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 Abstract Locate points

### Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using

Ch 13 - RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric

### PARAMETRIC EQUATIONS AND POLAR COORDINATES

10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. PARAMETRIC EQUATIONS

### MATH115. Polar Coordinate System and Polar Graphs. Paolo Lorenzo Bautista. June 14, De La Salle University

MATH115 Polar Coordinate System and Paolo Lorenzo Bautista De La Salle University June 14, 2014 PLBautista (DLSU) MATH115 June 14, 2014 1 / 30 Polar Coordinates and PLBautista (DLSU) MATH115 June 14, 2014

### Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of any angle,

### Section 10.1 Polar Coordinates

Section 10.1 Polar Coordinates Up until now, we have always graphed using the rectangular coordinate system (also called the Cartesian coordinate system). In this section we will learn about another system,

### 1) The domain of y = sin-1x is The range of y = sin-1x is. 2) The domain of y = cos-1x is The range of y = cos-1x is

MAT 204 NAME TEST 4 REVIEW ASSIGNMENT Sections 8.1, 8.3-8.5, 9.2-9.3, 10.1 For # 1-3, fill in the blank with the appropriate interval. 1) The domain of y = sin-1x is The range of y = sin-1x is 2) The domain

### Unit 7: Trigonometry Part 1

100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan(

### 5-2 Verifying Trigonometric Identities

5- Verifying Trigonometric Identities Verify each identity. 1. (sec 1) cos = sin 3. sin sin 3 = sin cos 4 5. = cot 7. = cot 9. + tan = sec Page 1 5- Verifying Trigonometric Identities 7. = cot 9. + tan

### Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46

Polar Coordinates Polar Coordinates: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ)

### MAC Learning Objectives. Module 12 Polar and Parametric Equations. Polar and Parametric Equations. There are two major topics in this module:

MAC 4 Module 2 Polar and Parametric Equations Learning Objectives Upon completing this module, you should be able to:. Use the polar coordinate system. 2. Graph polar equations. 3. Solve polar equations.

### sin30 = sin60 = cos30 = cos60 = tan30 = tan60 =

Precalculus Notes Trig-Day 1 x Right Triangle 5 How do we find the hypotenuse? 1 sinθ = cosθ = tanθ = Reciprocals: Hint: Every function pair has a co in it. sinθ = cscθ = sinθ = cscθ = cosθ = secθ = cosθ

### 10.7. Polar Coordinates. Introduction. What you should learn. Why you should learn it. Example 1. Plotting Points on the Polar Coordinate System

_7.qxd /8/5 9: AM Page 779 Section.7 Polar Coordinates 779.7 Polar Coordinates What ou should learn Plot points on the polar coordinate sstem. Convert points from rectangular to polar form and vice versa.

### Section 6.2 Graphs of the Other Trig Functions

Section 62 Graphs of the Other Trig Functions 369 Section 62 Graphs of the Other Trig Functions In this section, we will explore the graphs of the other four trigonometric functions We ll begin with the

### CLEP Pre-Calculus. Section 1: Time 30 Minutes 50 Questions. 1. According to the tables for f(x) and g(x) below, what is the value of [f + g]( 1)?

CLEP Pre-Calculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (non-cas) is allowed to be used for this section..

### Notice there are vertical asymptotes whenever y = sin x = 0 (such as x = 0).

1 of 7 10/1/2004 6.4 GRAPHS OF THE OTHER CIRCULAR 6.4 GRAPHS OF THE OTHER CIRCULAR Graphs of the Cosecant and Secant Functions Graphs of the Tangent and Cotangent Functions Addition of Ordinates Graphs

### Chapter 5.6: The Other Trig Functions

Chapter 5.6: The Other Trig Functions The other four trig functions, tangent, cotangent, cosecant, and secant are not sinusoids, although they are still periodic functions. Each of the graphs of these

### Polar Coordinates. Calculus 2 Lia Vas. If P = (x, y) is a point in the xy-plane and O denotes the origin, let

Calculus Lia Vas Polar Coordinates If P = (x, y) is a point in the xy-plane and O denotes the origin, let r denote the distance from the origin O to the point P = (x, y). Thus, x + y = r ; θ be the angle

### 1. GRAPHS OF THE SINE AND COSINE FUNCTIONS

GRAPHS OF THE CIRCULAR FUNCTIONS 1. GRAPHS OF THE SINE AND COSINE FUNCTIONS PERIODIC FUNCTION A period function is a function f such that f ( x) f ( x np) for every real numer x in the domain of f every

### Math 144 Activity #4 Connecting the unit circle to the graphs of the trig functions

144 p 1 Math 144 Activity #4 Connecting the unit circle to the graphs of the trig functions Graphing the sine function We are going to begin this activity with graphing the sine function ( y = sin x).

### PLANE TRIGONOMETRY Exam I September 13, 2007

Name Rec. Instr. Rec. Time PLANE TRIGONOMETRY Exam I September 13, 2007 Page 1 Page 2 Page 3 Page 4 TOTAL (10 pts.) (30 pts.) (30 pts.) (30 pts.) (100 pts.) Below you will find 10 problems, each worth

### 1. Be sure to complete the exploration before working on the rest of this worksheet.

PreCalculus Worksheet 4.1 1. Be sure to complete the exploration before working on the rest of this worksheet.. The following angles are given to you in radian measure. Without converting to degrees, draw

### Cumulative Review: SOHCAHTOA and Angles of Elevation and Depression

Cumulative Review: SOHCAHTOA and Angles of Elevation and Depression Part 1: Model Problems The purpose of this worksheet is to provide students the opportunity to review the following topics in right triangle

### The Sine and Cosine Functions

Concepts: Graphs of Tangent, Cotangent, Secant, and Cosecant. We obtain the graphs of the other trig functions by thinking about how they relate to the sin x and cos x. The Sine and Cosine Functions Page

### PARAMETERIZATIONS OF PLANE CURVES

PARAMETERIZATIONS OF PLANE CURVES Suppose we want to plot the path of a particle moving in a plane. This path looks like a curve, but we cannot plot it like we would plot any other type of curve in the

### MATHEMATICS 105 Plane Trigonometry

Chapter I THE TRIGONOMETRIC FUNCTIONS MATHEMATICS 105 Plane Trigonometry INTRODUCTION The word trigonometry literally means triangle measurement. It is concerned with the measurement of the parts, sides,

### 9.1 Parametric Curves

Math 172 Chapter 9A notes Page 1 of 20 9.1 Parametric Curves So far we have discussed equations in the form. Sometimes and are given as functions of a parameter. Example. Projectile Motion Sketch and axes,

### Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant Practice Problems. Questions

Questions 1. Describe the graph of the function in terms of basic trigonometric functions. Locate the vertical asymptotes and sketch two periods of the function. y = 3 tan(x/2) 2. Solve the equation csc

### DAY 1 - GEOMETRY FLASHBACK

DAY 1 - GEOMETRY FLASHBACK Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse sin θ = opp. hyp. cos θ = adj. hyp. tan θ = opp. adj. Tangent Opposite Adjacent a 2 + b 2 = c 2 csc θ = hyp. opp. sec θ =

### 4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch

### MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand. Overview

MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand Overview Below are the guidelines for constructing a graph of a trigonometric function

### Chapter 10 Homework: Parametric Equations and Polar Coordinates

Chapter 1 Homework: Parametric Equations and Polar Coordinates Name Homework 1.2 1. Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the

### Section 5.3 Graphs of the Cosecant and Secant Functions 1

Section 5.3 Graphs of the Cosecant, Secant, Tangent, and Cotangent Functions The Cosecant Graph RECALL: 1 csc x so where sin x 0, csc x has an asymptote. sin x To graph y Acsc( Bx C) D, first graph THE

### Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Approximate the coordinates of each turning point by graphing f(x) in the standard viewing

### The Polar Coordinate System

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-008 The Polar Coordinate System Alisa Favinger University

### Unit 2 Intro to Angles and Trigonometry

HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of

### MATH 1113 Exam 3 Review. Fall 2017

MATH 1113 Exam 3 Review Fall 2017 Topics Covered Section 4.1: Angles and Their Measure Section 4.2: Trigonometric Functions Defined on the Unit Circle Section 4.3: Right Triangle Geometry Section 4.4:

### PreCalculus Chapter 9 Practice Test Name:

This ellipse has foci 0,, and therefore has a vertical major axis. The standard form for an ellipse with a vertical major axis is: 1 Note: graphs of conic sections for problems 1 to 1 were made with the

### 12-3 Surface Areas of Pyramids and Cones

18. MOUNTAINS A conical mountain has a radius of 1.6 kilometers and a height of 0.5 kilometer. What is the lateral area of the mountain? The radius of the conical mountain is 1.6 kilometers and the height

### Chapter 3. Radian Measure and the Unit Circle. For exercises 23 28, answers may vary

Chapter Radian Measure and the Unit Circle Section....... 7. 8. 9. 0...... 7 8. 7. 0 8. 0 9. 0 0... 0 Radian Measure For exercises 8, answers may vary.. Multiply the degree measure by radian 80 and reduce.

### Vertical and Horizontal Translations

SECTION 4.3 Vertical and Horizontal Translations Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the vertical translation of a sine or cosine function. Find the horizontal

### 4.2 Graphing Inverse Trigonometric Functions

4.2 Graphing Inverse Trigonometric Functions Learning Objectives Understand the meaning of restricted domain as it applies to the inverses of the six trigonometric functions. Apply the domain, range and

### 8.6 Other Trigonometric Functions

8.6 Other Trigonometric Functions I have already discussed all the trigonometric functions and their relationship to the sine and cosine functions and the x and y coordinates on the unit circle, but let

### Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8)

Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Standards: Geom 19.0, Geom 20.0, Trig 7.0, Trig 8.0, Trig 12.0 Segerstrom High School -- Math Analysis Honors Name: Period:

### Chapter P: Preparation for Calculus

1. Which of the following is the correct graph of y = x x 3? E) Copyright Houghton Mifflin Company. All rights reserved. 1 . Which of the following is the correct graph of y = 3x x? E) Copyright Houghton

### Sec 4.1 Trigonometric Identities Basic Identities. Name: Reciprocal Identities:

Sec 4. Trigonometric Identities Basic Identities Name: Reciprocal Identities: Quotient Identities: sin csc cos sec csc sin sec cos sin tan cos cos cot sin tan cot cot tan Using the Reciprocal and Quotient

### This is called the horizontal displacement of also known as the phase shift.

sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for

### Inverses of Trigonometric. Who uses this? Hikers can use inverse trigonometric functions to navigate in the wilderness. (See Example 3.

1-4 Inverses of Trigonometric Functions Objectives Evaluate inverse trigonometric functions. Use trigonometric equations and inverse trigonometric functions to solve problems. Vocabulary inverse sine function

### 8B.2: Graphs of Cosecant and Secant

Opp. Name: Date: Period: 8B.: Graphs of Cosecant and Secant Or final two trigonometric functions to graph are cosecant and secant. Remember that So, we predict that there is a close relationship between

### Foundations of Math II

Foundations of Math II Unit 6b: Toolkit Functions Academics High School Mathematics 6.6 Warm Up: Review Graphing Linear, Exponential, and Quadratic Functions 2 6.6 Lesson Handout: Linear, Exponential,

### Unit 4. Applications of integration

Unit 4. Applications of integration 4A. Areas between curves. 4A-1 Find the area between the following curves a) y = 2x 2 and y = 3x 1 b) y = x 3 and y = ax; assume a > 0 c) y = x + 1/x and y = 5/2. d)

### You are not expected to transform y = tan(x) or solve problems that involve the tangent function.

In this unit, we will develop the graphs for y = sin(x), y = cos(x), and later y = tan(x), and identify the characteristic features of each. Transformations of y = sin(x) and y = cos(x) are performed and

### Chapter 4. Trigonometric Functions. 4.6 Graphs of Other. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 4 Trigonometric Functions 4.6 Graphs of Other Trigonometric Functions Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Understand the graph of y = tan x. Graph variations of y =

### Lesson 5.6: Angles in Standard Position

Lesson 5.6: Angles in Standard Position IM3 - Santowski IM3 - Santowski 1 Fast Five Opening Exercises! Use your TI 84 calculator:! Evaluate sin(50 ) " illustrate with a diagram! Evaluate sin(130 ) " Q

### Math Handbook of Formulas, Processes and Tricks. Trigonometry

Math Handbook of Formulas, Processes and Tricks (www.mathguy.us) Trigonometry Prepared by: Earl L. Whitney, FSA, MAAA Version 2.1 April 10, 2017 Copyright 2012 2017, Earl Whitney, Reno NV. All Rights Reserved

### Section Polar Coordinates. or 4 π (restricting θ to the domain of the lemniscate). So, there are horizontal tangents at ( 4 3

Section 10.3 Polar Coordinates 66. r = e θ x = r cos θ = e θ cos θ, y = r sin θ = e θ sin θ. = eθ sin θ+e θ cos θ = e θ (sin θ+cos θ), dx = eθ cos θ e θ sin θ = e θ (cos θ sin θ). Let 1 = 0 sin θ = cos

### Math 1201 Chapter 2 Review

ath 1201 hapter 2 Review ultiple hoice Identify the choice that best completes the statement or answers the question. 1. etermine tan and tan. 8 10 a. tan = 1.25; tan = 0.8 c. tan = 0.8; tan = 1.25 b.

Vectors Some physical quantities have both size and direction. These physical quantities are represented with vectors. A common example of a physical quantity that is represented with a vector is a force.

### Nelson Functions 11 Errata

Nelson Functions 11 Errata 1: Introduction to Functions Location Question Correct Answer Getting Started 6a Graph is correct but vertex and axis of symmetry are not labelled. Add blue point labelled (in

### Unit 4 Graphs of Trigonometric Functions - Classwork

Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x ( decimal places) on the chart and graph the points on the graph below. x 0 o 30

### Section 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.

1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint

### The equation of the axis of symmetry is. Therefore, the x-coordinate of the vertex is 2.

1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. Here, a = 2, b = 8, and c

### Computer Graphics : Bresenham Line Drawing Algorithm, Circle Drawing & Polygon Filling

Computer Graphics : Bresenham Line Drawing Algorithm, Circle Drawing & Polygon Filling Downloaded from :www.comp.dit.ie/bmacnamee/materials/graphics/006- Contents In today s lecture we ll have a loo at:

### Section 7.6 Graphs of the Sine and Cosine Functions

Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xy-plane. Just like with any other function, it is easy to do by plotting points.

### 9-1 GCSE Maths. GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9).

9-1 GCSE Maths GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9). In each tier, there are three exams taken at the end of Year 11. Any topic may be assessed on each of

### 4. The following diagram shows the triangle AOP, where OP = 2 cm, AP = 4 cm and AO = 3 cm.

Circular Functions and Trig - Practice Problems (to 07) 1. In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm. Calculate (a) the size of ; (b) the area of triangle PQR. 2. The following diagram shows

### Solving Right Triangles. SECURITY A security light is being

5-5 OJECTIVES Evaluate inverse trigonometric functions. Find missing angle measurements. Solve right triangles. Solving Right Triangles SECURITY A security light is being installed outside a loading dock.

### Unit 4 Graphs of Trigonometric Functions - Classwork

Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x decimal places) on the chart and graph the points on the graph below. x 0 o 30 o

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

1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for f (x) = 2x 2 + 8x 3. Then graph the function by making a table of values. 2. Determine whether f (x)

### SENIOR HIGH MATH LEAGUE April 24, GROUP IV Emphasis on TRIGONOMETRY

SENIOR HIGH MATH LEAGUE TEST A Write all radical expressions in simplified form and unless otherwise stated give exact answers. 1. Give the exact value for each of the following where the angle is given

### Trigonometry. Secondary Mathematics 3 Page 180 Jordan School District

Trigonometry Secondary Mathematics Page 80 Jordan School District Unit Cluster (GSRT9): Area of a Triangle Cluster : Apply trigonometry to general triangles Derive the formula for the area of a triangle

### Algebra II. Chapter 13 Notes Sections 13.1 & 13.2

Algebra II Chapter 13 Notes Sections 13.1 & 13.2 Name Algebra II 13.1 Right Triangle Trigonometry Day One Today I am using SOHCAHTOA and special right triangle to solve trig problems. I am successful

### You can take the arccos of both sides to get θ by itself.

.7 SOLVING TRIG EQUATIONS Example on p. 8 How do you solve cos ½ for? You can tae the arccos of both sides to get by itself. cos - (cos ) cos - ( ½) / However, arccos only gives us an answer between 0

### 1-7 Inverse Relations and Functions

Graph each function using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no. 1. f (x) = x 2 + 6x + 9 The graph of f (x) = x 2 +

### Bailey Kirkland Education Group, LLC Common Core State Standard I Can Statements 4 th Grade Mathematics 6/18/2013

Bailey Kirkland Education Group, LLC Common Core State Standard 4 th Grade Mathematics 6/18/2013 CCSS Key: Operations and Algebraic Thinking (OA) Number and Operations in Base Ten (NBT) Numbers and Operations

### 3.1 The Inverse Sine, Cosine, and Tangent Functions

3.1 The Inverse Sine, Cosine, and Tangent Functions Let s look at f(x) = sin x The domain is all real numbers (which will represent angles). The range is the set of real numbers where -1 sin x 1. However,

### Geometry: Chapter 7. Name: Class: Date: 1. Find the length of the leg of this right triangle. Give an approximation to 3 decimal places.

Name: Class: Date: Geometry: Chapter 7 1. Find the length of the leg of this right triangle. Give an approximation to 3 decimal places. a. 12.329 c. 12.650 b. 11.916 d. 27.019 2. ABC is a right triangle.

### Let be a function. We say, is a plane curve given by the. Let a curve be given by function where is differentiable with continuous.

Module 8 : Applications of Integration - II Lecture 22 : Arc Length of a Plane Curve [Section 221] Objectives In this section you will learn the following : How to find the length of a plane curve 221

### Unit 5 Day 5: Law of Sines and the Ambiguous Case

Unit 5 Day 5: Law of Sines and the Ambiguous Case Warm Up: Day 5 Draw a picture and solve. Label the picture with numbers and words including the angle of elevation/depression and height/length. 1. The

### Function f. Function f -1

Page 1 REVIEW (1.7) What is an inverse function? Do all functions have inverses? An inverse function, f -1, is a kind of undoing function. If the initial function, f, takes the element a to the element

### Pre-calculus Chapter 4 Part 1 NAME: P.

Pre-calculus NAME: P. Date Day Lesson Assigned Due 2/12 Tuesday 4.3 Pg. 284: Vocab: 1-3. Ex: 1, 2, 7-13, 27-32, 43, 44, 47 a-c, 57, 58, 63-66 (degrees only), 69, 72, 74, 75, 78, 79, 81, 82, 86, 90, 94,

### Proof of Identities TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System

Math Objectives Students will be able to interpret reciprocal, negative angle, cofunction, and Pythagorean identities in terms of the graphs of the trigonometric functions involved Students will be able

### Unit 4 Graphs of Trigonometric Functions - Classwork

Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x (2 decimal places) on the chart and graph the points on the graph below. x 0 o 30

### 10 Polar Coordinates, Parametric Equations

Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

### MATHS. years 4,5,6. malmesbury c of e primary school NAME CLASS

MATHS years 4,5,6 NAME CLASS malmesbury c of e primary school LEARNING LADDERS CONTENTS Ladder Title Times Tables Addition Subtraction Multiplication Division Fractions Decimals Percentage and Ratio Problem

### 12 Polar Coordinates, Parametric Equations

54 Chapter Polar Coordinates, Parametric Equations Polar Coordinates, Parametric Equations Just as we describe curves in the plane using equations involving x and y, so can we describe curves using equations

### UNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction

Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system

### 10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: 2. SOLUTION: 3.

Find x. Assume that segments that appear to be tangent are tangent. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. 5. SCIENCE A piece of broken pottery found at an archaeological site is shown.

### 1-5 Parent Functions and Transformations

Describe the following characteristics of the graph of each parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. 1.

### 5.2. The Sine Function and the Cosine Function. Investigate A

5.2 The Sine Function and the Cosine Function What do an oceanographer, a stock analyst, an audio engineer, and a musician playing electronic instruments have in common? They all deal with periodic patterns.

### Name (s) Class Date ERROR ANALYSIS GEOMETRY WORD PROBLEMS

7 th Grade Common Core Name (s) Class Date ERROR ANALYSIS GEOMETRY WORD PROBLEMS Includes: * Angles * Triangles * Scale Drawings * Area and Circumference of a Circle * Volume of Prisms and Pyramids * Surface

### turn counterclockwise from the positive x-axis. However, we could equally well get to this point by a 3 4 turn clockwise, giving (r, θ) = (1, 3π 2

Math 133 Polar Coordinates Stewart 10.3/I,II Points in polar coordinates. The first and greatest achievement of modern mathematics was Descartes description of geometric objects b numbers, using a sstem

### Proving Trigonometric Identities

MHF 4UI Unit 7 Day Proving Trigonometric Identities An identity is an epression which is true for all values in the domain. Reciprocal Identities csc θ sin θ sec θ cos θ cot θ tan θ Quotient Identities

### 32 ft. 48 ft. 15 cups 36, 60, 84

2012 2013 Geometry / Geometry Honors Second Semester Study Guide 1. Matthew is 4 feet tall. t 5 o clock in the afternoon, Matthew casts a shadow 20 feet long. He is standing net to a telephone pole that