Semester 2, Unit 4: Activity 21

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Semester 2, Unit 4: Activity 21"

Transcription

1 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 21 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 21-1: Define the reciprocal and quotient identities. Use and transform the Pythagorean Identity. Example Lesson 21-1: Page 1 of 21

2 Page 2 of 21

3 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 22 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 22-1 and 22-2: Use the unit circle to write equivalent trigonometric equations. Write cofunction identities for Sine and Cosine. Solve trigonometric equations using identities and by graphing. Example Lesson 22-1: Math Tip: To convert degrees to radians, multiply the degree measure by Page 3 of 21

4 Example Lesson 22-2: Page 4 of 21

5 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 23 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 23-1, 23-2, and 23-3: Use Sum and Difference Identities. Use Half Angle Identity. Derive the identities and use them to find the exact values of trigonometric functions. Use trigonometric identities to solve equations. Example Lesson 23-1: Page 5 of 21

6 Page 6 of 21

7 Example Lesson 23-2: Sum and Difference Identities Page 7 of 21

8 Page 8 of 21

9 Example Lesson 23-3: Page 9 of 21

10 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 24 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 24-2: Write equations for the Law of Cosines using a standard angle. Apply Law of Cosines in a real world problem. Example Lesson 24-2: Law of Cosines. The Law of Cosines is useful in many applications involving non-right triangles, also known as oblique triangles. Page 10 of 21

11 Page 11 of 21

12 Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities Law of Cosines Oblique Triangle Law of Sines Ambiguous Case (SSA) PreCalculus Honors: Functions and Their Graphs Semester 2, Unit 4: Activity 25 Unit Overview In this unit students will extend their knowledge of trigonometry as they study trigonometric identities, equations, and formulas. Students will explore the Law of Cosines and the Law of Sines, and apply them to solve non-right triangles. Student Focus Main Ideas for success in lessons 25-1 and 25-2: Discover mathematical relations to derive the Law of Sines. Find unknown sides and angles of an oblique triangle by using the Law of Sines. Example Lesson 25-1: Like the Law of Cosines, the Law of Sines relates the sides and angles in an oblique triangle, and these can be used to find unknown sides or angles given at least three known measures that are not all angle measures. Law of Sines Page 12 of 21

13 Page 13 of 21 Example

14 Page 14 of 21 Lesson 25-2:

15 Page 15 of 21

16 NAME CLASS DATE LEssON Make use of structure. Express each trigonometric function as a reciprocal to write an identity. u a. sec 2 Precalculus Unit 4 Practice 5. Make sense of problems. Suppose you know that cos u a. Using the Pythagorean identity, what can you conclude about sin u? ( ) b. tan x 12p b. How does your answer to part a change if you know that u lies in Quadrant III? 2. Make use of structure. Express each trigonometric function as a quotient to write an identity. 2p a. cot 3 b. tan (x 2 308) 3. Make use of structure. Express each trigonometric function using the Pythagorean Theorem to write an identity. 2 p a. tan c. What is the approximate value of u, in degrees, given that u lies in Quadrant III? LEssON Which expression is equivalent to tan u csc u? A. B. cosu sin 2 u 1 sec u C. csc u D. sec u b. csc 2 (x ) 2 sec t 1 7. Reason abstractly. Simplify sec 22. t 4. Consider the function f( x) 5sin 2 x1 cos 2 x. What is 23 f 181? A B C D College Board. All rights reserved. 1 SpringBoard Precalculus, Unit 4 Practice Page 16 of 21

17 NAME CLASS DATE Attend to precision. For Items 8 and 9, verify each identity. sin x 8. 5cosx tanx 13. Reason abstractly. Use the unit circle to find the value of each of the following. P(c, d ) 1 9. (1 2 cos 2 a)(1 1 cot 2 a) Simone attempted to verify the identity sin 2 u csc u sec u 5 tan u by entering sin 2 u csc u sec u 2 tan u as function Y 1 in the graphing calculator. Is her method correct? Explain. a. cos (2p 2 u) b. sin ( u) LEssON 22-1 For Items 11 and 12, complete each statement. p 11. If cos x 5 0.8, then sin x A. 0.8 B C D p 14. Given that sin ى 0.34, 9 name three additional angles whose sines are approximately equal to Verify the identity cos ( u) tan u 5 2sin u. Be sure to justify each step of your argument with a valid reason. 12. Make use of structure. If tan x 5 1.2, then tan (p 2 x) College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 17 of 21

18 NAME CLASS DATE LEssON Attend to precision. Solve each equation over the given interval without a calculator. 1 a. 2 x2 x5 2 cot cos 0, p, 32 p b. 4 tan 2 a 5 sin 2 a, 0, p 2 LEssON 23-1 The sound of a single musical note can be represented using the function y 5 a sin (2p ft), where a is the amplitude (volume) of the sound measured in decibels (db), f is the frequency (pitch) of the note measured in hertz (Hz), and t is time. 17. What is the solution of cos 2 x over the interval [08, 3608)? A. 58 B C. infinite number of solutions D. no solution 21. Model with mathematics. The frequency of the musical note middle G is about 208 Hz. Write the equation for the sine wave that represents middle G played at a volume of 55 db. 22. Write the equation for the sine wave for the note one octave below middle G played at a volume of 80 db. 18. Use appropriate tools strategically. Solve the equation over the given interval. You may use a calculator. 3 tan 2 u , [08, 3608) 19. Evan solved the equation cos x 5 21 sin x on the interval p 2, p. His work is shown below. Explain his error and find the correct solution. 23. Make use of structure. Write the equation for the sine wave that represents the sound when both notes in Items 21 and 22 are played at the same time. 24. Use a calculator to graph the functions = ٣, y 1 ى ٢ x ١ 2 cos 2 52, and y 1 1 y 2 on the interval [22p, 2p], and then make a sketch in the space below. Step 1: cos x 5 21 sin x Original equation cos x Step 2: 521 sin x Divide both sides by cos x. Step 3: tan x 5 21 Definition of tangent 3p Step 4: x 5 4 Solve for x. 20. Explain how the answer to tan 2 x 51, [08, 3608) could not be an infinite number of solutions. 25. Express the function y 1 1 y 2 in the form y 5 sin (x 1 c) College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 18 of 21

19 NAME CLASS DATE LEssON Make sense of problems. In Item 25, you found the function y 1 1 y 2 in the form y 5 sin (x 1 c). Use the sum identity for sine to algebraically verify this identity. LEssON 23-3 Make use of structure. Verify each identity. 31. sin (x 2 p) 5 2sin x 27. Make use of structure. Find the exact value. a. sin 15 p 6 b. tan (1058) ( ) cos u 1 a tanu tan a 5 cosu cos a Given sin a5 and tan b52 with angle 5 10 a terminating in Quadrant I and angle b terminating in Quadrant II, what is the exact value of sin (a 1 b)? A. 7 5 B. C D Attend to precision. Solve each equation on the interval [08, 3608]. 33. sin 2u 2 cos u cos 4 u 2 sin 4 u 5 cos 2u Given sin A5 and tan B52 with /A 5 10 terminating in Quadrant I and /B terminating in Quadrant II, find the exact value of tan (A2B) Given cot θ5 6 with 0 < u ڤ p 2 sin 2u, and tan 2u., find sin u, cos u, 35. Determine the number of solutions on the interval [0, 2p ) for cos (4u) A. 2 B. 4 C. 6 D College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 19 of 21

20 NAME CLASS DATE LEssON 24-1 For Items 36 40, refer to Item 17a in Lesson Make sense of problems. Explain why the speed of the blade is at 0 ft/sec at the value(s) you found. Model with mathematics. For Items 37 40, find the distance from the center of the wheel to the stirrer blade for each angle From Ghirardelli Square in San Francisco, you can see the Golden Gate Bridge and Alcatraz Island. The angle between the sight lines to these landmarks is approximately 808. The approximate distance from Ghirardelli Square to the Golden Gate Bridge is 3.2 miles and to Alcatraz is 1.4 miles. A surveyor precisely measured the angle between the sight lines to be By how many miles does the approximate distance change from the Golden Gate Bridge to Alcatraz? At which value of u does the speed of the blade reach a maximum? 43. The sides of an isosceles triangle have lengths of 7.2, 7.2, and What are the angles of the triangle? A. 778 B C D A triangle has side lengths of 8, 10, and 12. What are the angles of the triangle? LEssON Model with mathematics. A new courtyard at Ghirardelli Square in San Francisco will be triangular, as shown in the diagram below. Retaining Wall 45 yd 30 yd 1058 Suppose the angle opposite of the retaining wall needs to be decreased by 78. If the lengths of the courtyard are to remain the same, find the length of the retaining wall and the new angle measurement. 45. A triangle has two side lengths of 1 2 and 2 and an 3 angle measure of 328 between them. Which of the following is closest to the length of the remaining side? 1 A B C D College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 20 of 21

21 NAME CLASS DATE LEssON 25-1 For Items 46 48, refer to Items 1 and 2 in Lesson Model with mathematics. Suppose the plane traveled on its diverted path instead of turning course towards Honolulu after 1.5 hours. a. How many more miles would the plane have traveled after 2 hours? LEssON Determine how many triangles are possible with the given information: 428, b 5 60, a A. no triangle B. one triangle C. two triangles D. three triangles b. How far would the plane have been from Honolulu after 2 hours? 47. Which equation could be used to find the bearing of the plane if it had traveled for 2 hours and needed to turn at that point towards Honolulu? sin x sin20 A sin x sin20 B sin x sin20 C sin x sin20 D Construct viable arguments. Determine how many triangles are possible with the given information. Draw a sketch and show any calculations you used. a , b , a b. 60, b55 3, a Make use of structure. Solve the two-solution ambiguous case situation given b 5 20, a 5 25, B Attend to precision. Find the new bearing of the plane when it turns towards Honolulu after 2 hours. 49. In triangle DEF, angle D is 428, angle E is 788, and DE is Find angle F, EF, and DF. For Items 54 55, solve each triangle using the Law of Sines. 54. A 5 538, B 5 658, c In triangle STU, ST is 25, TU is 30, and SU is Find all three angles. 55. B 5 298, C 5 158, b College Board. All rights reserved. SpringBoard Precalculus, Unit 4 Practice Page 21 of 21

Analytic Trigonometry

Analytic Trigonometry Name Chapter 5 Analytic Trigonometry Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions and

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Right Triangle Trigonometry MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of acute angles, use

More information

Unit 6 Trigonometric Identities, Equations, and Applications

Unit 6 Trigonometric Identities, Equations, and Applications Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean

More information

SUM AND DIFFERENCE FORMULAS

SUM AND DIFFERENCE FORMULAS SUM AND DIFFERENCE FORMULAS Introduction We have several identities that we are concentrating on in this section: o Difference Identities for Cosine o Sum Identities for Cosine o Cofunction Identities

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

Solutions to Exercises, Section 5.1

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

More information

Trigonometry Lesson Objectives

Trigonometry Lesson Objectives Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

More information

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out: First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin

More information

M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.

M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity. M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides

More information

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)

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

More information

Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)

Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.) 1- The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview

More information

11 Trigonometric Functions of Acute Angles

11 Trigonometric Functions of Acute Angles Arkansas Tech University MATH 10: Trigonometry Dr. Marcel B. Finan 11 Trigonometric Functions of Acute Angles In this section you will learn (1) how to find the trigonometric functions using right triangles,

More information

Trigonometry. Week 1 Right Triangle Trigonometry

Trigonometry. Week 1 Right Triangle Trigonometry Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge

More information

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

More information

Section 9.4 Trigonometric Functions of any Angle

Section 9.4 Trigonometric Functions of any Angle Section 9. Trigonometric Functions of any Angle So far we have only really looked at trigonometric functions of acute (less than 90º) angles. We would like to be able to find the trigonometric functions

More information

Trigonometric Functions and Triangles

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

More information

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes 210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:

More information

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places. SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.

More information

Inverse Trigonometric Functions

Inverse Trigonometric Functions SECTION 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the exact value of an inverse trigonometric function. Use a calculator to approximate

More information

Trigonometric Identities and Conditional Equations C

Trigonometric Identities and Conditional Equations C Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving real-world problems and in the development of mathematics. Whatever their use, it is often of value

More information

MPE Review Section II: Trigonometry

MPE Review Section II: Trigonometry MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the

More information

1. Introduction identity algbriac factoring identities

1. Introduction identity algbriac factoring identities 1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

More information

Verifying Trigonometric Identities. Introduction. is true for all real numbers x. So, it is an identity. Verifying Trigonometric Identities

Verifying Trigonometric Identities. Introduction. is true for all real numbers x. So, it is an identity. Verifying Trigonometric Identities 333202_0502.qxd 382 2/5/05 Chapter 5 5.2 9:0 AM Page 382 Analytic Trigonometry Verifying Trigonometric Identities What you should learn Verify trigonometric identities. Why you should learn it You can

More information

Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6.

Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6. 1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive,

More information

6.6 The Inverse Trigonometric Functions. Outline

6.6 The Inverse Trigonometric Functions. Outline 6.6 The Inverse Trigonometric Functions Tom Lewis Fall Semester 2015 Outline The inverse sine function The inverse cosine function The inverse tangent function The other inverse trig functions Miscellaneous

More information

CRASH COURSE IN PRECALCULUS

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

More information

Chapter 5: Trigonometric Functions of Real Numbers

Chapter 5: Trigonometric Functions of Real Numbers Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the

More information

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

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

More information

18 Verifying Trigonometric Identities

18 Verifying Trigonometric Identities Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 18 Verifying Trigonometric Identities In this section, you will learn how to use trigonometric identities to simplify trigonometric

More information

Trigonometric Functions: The Unit Circle

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

More information

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions Section 8.1: The Inverse Sine, Cosine, and Tangent Functions The function y = sin x doesn t pass the horizontal line test, so it doesn t have an inverse for every real number. But if we restrict the function

More information

PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

More information

TRIGONOMETRIC EQUATIONS

TRIGONOMETRIC EQUATIONS CHAPTER 3 CHAPTER TABLE OF CONTENTS 3- First-Degree Trigonometric Equations 3- Using Factoring to Solve Trigonometric Equations 3-3 Using the Quadratic Formula to Solve Trigonometric Equations 3-4 Using

More information

Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179

Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179 Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.

More information

Section 6-3 Double-Angle and Half-Angle Identities

Section 6-3 Double-Angle and Half-Angle Identities 6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

More information

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.

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

More information

Trigonometric Functions

Trigonometric Functions Trigonometric Functions MATH 10, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: identify a unit circle and describe its relationship to real

More information

4.1 Radian and Degree Measure

4.1 Radian and Degree Measure Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Assume that the cities lie on the same north-south line and that the radius of the earth

More information

About Trigonometry. Triangles

About Trigonometry. Triangles About Trigonometry TABLE OF CONTENTS About Trigonometry... 1 What is TRIGONOMETRY?... 1 Triangles... 1 Background... 1 Trigonometry with Triangles... 1 Circles... 2 Trigonometry with Circles... 2 Rules/Conversion...

More information

Inverse Circular Function and Trigonometric Equation

Inverse Circular Function and Trigonometric Equation Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse

More information

Section 6-4 Product Sum and Sum Product Identities

Section 6-4 Product Sum and Sum Product Identities 480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 6-4 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing

More information

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 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

More information

MTH 122 Plane Trigonometry Fall 2015 Test 1

MTH 122 Plane Trigonometry Fall 2015 Test 1 MTH 122 Plane Trigonometry Fall 2015 Test 1 Name Write your solutions in a clear and precise manner. Answer all questions. 1. (10 pts) a). Convert 44 19 32 to degrees and round to 4 decimal places. b).

More information

Enrichment The Physics of Soccer Recall from Lesson 7-1 that the formula for the maximum height h h v 0 2 sin 2

Enrichment The Physics of Soccer Recall from Lesson 7-1 that the formula for the maximum height h h v 0 2 sin 2 7-1 The Physics of Soccer Recall from Lesson 7-1 that the formula for the maximum height h h v 0 2 sin 2 of a projectile is 2g, where is the measure of the angle of elevation in degrees, v 0 is the initial

More information

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY Right Triangle Trigonometry Page 1 of 15 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

More information

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

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

More information

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

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

More information

Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

More information

Angles and Their Measure

Angles and Their Measure Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150 Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. ) 56

More information

Trigonometry Hard Problems

Trigonometry Hard Problems Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.

More information

pp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64

pp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64 Semester 1 Text: Chapter 1: Tools of Algebra Lesson 1-1: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 1-2: Algebraic Expressions

More information

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1 Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

More information

29 Wyner PreCalculus Fall 2016

29 Wyner PreCalculus Fall 2016 9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves

More information

6.1 Basic Right Triangle Trigonometry

6.1 Basic Right Triangle Trigonometry 6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at

More information

Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

More information

5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved.

5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved. 5.2 Trigonometric Functions Of Real Numbers Copyright Cengage Learning. All rights reserved. Objectives The Trigonometric Functions Values of the Trigonometric Functions Fundamental Identities 2 Trigonometric

More information

Chapter 6 Trigonometric Functions of Angles

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.

More information

Math Placement Test Practice Problems

Math Placement Test Practice Problems Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

More information

Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 42. The sum of the measures of the angles of a triangle is 180. Therefore, The sine of an angle

More information

Overview. Essential Questions. Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities. Number of instruction days: 5 7 (1 day = 53 minutes)

Overview. Essential Questions. Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities. Number of instruction days: 5 7 (1 day = 53 minutes) Precalculus, Quarter, Unit.5 Proving Trigonometric Identities Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Verify proofs of Pythagorean identities. Apply Pythagorean,

More information

Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.

Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem. Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

Trigonometric Identities

Trigonometric Identities Trigonometric Identities Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

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

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

More information

6.3 Inverse Trigonometric Functions

6.3 Inverse Trigonometric Functions Chapter 6 Periodic Functions 863 6.3 Inverse Trigonometric Functions In this section, you will: Learning Objectives 6.3.1 Understand and use the inverse sine, cosine, and tangent functions. 6.3. Find the

More information

Graphing Trigonometric Skills

Graphing Trigonometric Skills Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

More information

Trigonometry LESSON TWO - The Unit Circle Lesson Notes

Trigonometry LESSON TWO - The Unit Circle Lesson Notes (cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each

More information

Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring

Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest

More information

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

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

More information

4-1 Right Triangle Trigonometry

4-1 Right Triangle Trigonometry Find the exact values of the six trigonometric functions of θ. 1. The length of the side opposite θ is 8 is 18., the length of the side adjacent to θ is 14, and the length of the hypotenuse 3. The length

More information

Plane Trigonometry - Fall 1996 Test File

Plane Trigonometry - Fall 1996 Test File Plane Trigonometry - Fall 1996 Test File Test #1 1.) Fill in the blanks in the two tables with the EXACT values of the given trigonometric functions. The total point value for the tables is 10 points.

More information

Name Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles

Name Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use

More information

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

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

More information

Able Enrichment Centre - Prep Level Curriculum

Able Enrichment Centre - Prep Level Curriculum Able Enrichment Centre - Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,

More information

Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

More information

1) Convert 13 32' 47" to decimal degrees. Round your answer to four decimal places.

1) Convert 13 32' 47 to decimal degrees. Round your answer to four decimal places. PRECLCULUS FINL EXM, PRCTICE UNIT ONE TRIGONOMETRIC FUNCTIONS ) Convert ' 47" to decimal degrees. Round your answer to four decimal places. ) Convert 5.6875 to degrees, minutes, and seconds. Round to the

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH0000 SEMESTER 1 016/017 DR. ANTHONY BROWN 5. Trigonometry 5.1. Parity and co-function identities. In Section 4.6 of Chapter 4 we looked

More information

5.2 Unit Circle: Sine and Cosine Functions

5.2 Unit Circle: Sine and Cosine Functions Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and

More information

Algebra 2/ Trigonometry Extended Scope and Sequence (revised )

Algebra 2/ Trigonometry Extended Scope and Sequence (revised ) Algebra 2/ Trigonometry Extended Scope and Sequence (revised 2012 2013) Unit 1: Operations with Radicals and Complex Numbers 9 days 1. Operations with radicals (p.88, 94, 98, 101) a. Simplifying radicals

More information

MATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions

MATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions Math 50 T9a-Right Triangle Trigonometry Review Page MTH 50 TOPIC 9 RIGHT TRINGLE TRIGONOMETRY 9a. Right Triangle Definitions of the Trigonometric Functions 9a. Practice Problems 9b. 5 5 90 and 0 60 90

More information

3. Right Triangle Trigonometry

3. Right Triangle Trigonometry . Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle

More information

Section 3.1 Radian Measure

Section 3.1 Radian Measure Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle

More information

Any two right triangles, with one other angle congruent, are similar by AA Similarity. This means that their side lengths are.

Any two right triangles, with one other angle congruent, are similar by AA Similarity. This means that their side lengths are. Lesson 1 Trigonometric Functions 1. I CAN state the trig ratios of a right triangle 2. I CAN explain why any right triangle yields the same trig values 3. I CAN explain the relationship of sine and cosine

More information

Evaluating trigonometric functions

Evaluating trigonometric functions MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned

More information

FURTHER TRIGONOMETRIC IDENTITIES AND EQUATIONS

FURTHER TRIGONOMETRIC IDENTITIES AND EQUATIONS Mathematics Revision Guides Further Trigonometric Identities and Equations Page of 7 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C4 Edexcel: C3 OCR: C3 OCR MEI: C4 FURTHER TRIGONOMETRIC

More information

39 Verifying Trigonometric Identities

39 Verifying Trigonometric Identities 39 Verifying Trigonometric Identities In this section, you will learn how to use trigonometric identities to simplify trigonometric expressions. Equations such as (x 2)(x + 2) x 2 4 or x2 x x + are referred

More information

Exact Values of the Sine and Cosine Functions in Increments of 3 degrees

Exact Values of the Sine and Cosine Functions in Increments of 3 degrees Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms

More information

Types of Angles acute right obtuse straight Types of Triangles acute right obtuse hypotenuse legs

Types of Angles acute right obtuse straight Types of Triangles acute right obtuse hypotenuse legs MTH 065 Class Notes Lecture 18 (4.5 and 4.6) Lesson 4.5: Triangles and the Pythagorean Theorem Types of Triangles Triangles can be classified either by their sides or by their angles. Types of Angles An

More information

Geometry Mathematics Curriculum Guide Unit 6 Trig & Spec. Right Triangles 2016 2017

Geometry Mathematics Curriculum Guide Unit 6 Trig & Spec. Right Triangles 2016 2017 Unit 6: Trigonometry and Special Right Time Frame: 14 Days Primary Focus This topic extends the idea of triangle similarity to indirect measurements. Students develop properties of special right triangles,

More information

Unit 8 Inverse Trig & Polar Form of Complex Nums.

Unit 8 Inverse Trig & Polar Form of Complex Nums. HARTFIELD PRECALCULUS UNIT 8 NOTES PAGE 1 Unit 8 Inverse Trig & Polar Form of Complex Nums. This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit. () Inverse Functions (3) Invertibility of Trigonometric

More information

Basic Trigonometry, Significant Figures, and Rounding - A Quick Review

Basic Trigonometry, Significant Figures, and Rounding - A Quick Review Basic Trigonometry, Significant Figures, and Rounding - A Quick Review (Free of Charge and Not for Credit) by Professor Patrick L. Glon, P.E. Basic Trigonometry, Significant Figures, and Rounding A Quick

More information

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles 4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred

More information