Squares on a Triangle
|
|
- Hilda Hamilton
- 6 years ago
- Views:
Transcription
1 Squares on a Triangle NAME The Pythagorean theorem states that the sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse of the right triangle. This is the basis for the formula a 2 + b 2 = c 2, since a 2, b 2, and c 2 are the areas of the actual squares on the sides of the triangle. But what if the triangle is not a right triangle? Open the Squares on a Triangle applet: Go to Illuminations,. Enter the Activities section, and search for squares. Select the Squares on a Triangle applet from the list of search results. 1. Compare the sum of Areas I and II to Area III for different measures of C. Drag the point in the slider to change the measure of C. Points A and B can also be dragged to change the size of the triangle. Record several observations, being sure to test cases for acute, right, and obtuse angle measures of C. MEASURE OF ANGLE C AREA SQUARE I + AREA SQUARE II AREA SQUARE III HOW DO THE AREAS COMPARE? WHAT TYPE OF TRIANGLE IS IT? (ACUTE, OBTUSE, RIGHT) 60º 70º 80º 90º 100º 110º 2. Use your observations to complete the following statements: In an acute triangle, if a, b, and c are the sides of a triangle and c is the longest, then a 2 + b 2 will be. In an obtuse triangle, if a, b, and c are the sides of a triangle and c is the longest, then a 2 + b 2 will be.
2 Use these two new ideas to answer the following questions. 3. Suppose a triangle has sides of lengths 9, 15, and 17. Is it an acute, obtuse, or a right triangle? 4. Suppose a triangle has sides of lengths 7, 24, and 25. Is it an acute, obtuse, or right triangle? 5. Suppose a triangle has sides of lengths 6, 14, and 16. Is it an acute, obtuse, or right triangle?
3 Defects in a Triangle You already know the Pythagorean theorem, which is an area relationship among the sides of a right triangle. Wouldn t it be nice if there were an area relationship for any triangle? Actually, there is a nice area relationship among the sides of any triangle. In this next section, you will look for this relationship. Click the Show Defect I button on the applet. You should see something similar to this picture: A similar result will occur when you click the Show Defect II button. Find a way to relate the Areas of Squares I and II and the two Defect Areas to Area III. You may wish to divide your work into cases, looking at acute triangles first and then obtuse triangles. The table below will help to organize your ideas. MEASURE OF ANGLE C AREA SQUARE I + AREA SQUARE II AREA DEFECT I AREA DEFECT II AREA SQUARE III WHAT TYPE OF TRIANGLE IS IT? (ACUTE OR OBTUSE) 60º 70º 80º 90º 100º 110º 120º
4 6. Briefly summarize what you discovered. Calculating Area Defects The relationships you found for the areas of the squares and defects of the sides of a triangle is called the law of cosines. You may be wondering what trigonometry has do with areas and defects, so let s examine this idea further. 7. Why is the defect area in this diagram equal to a x? 8. Sides b and x are part of a right triangle. What trigonometric function would relate them? (Write the trigonometric equation.) 9. How would you calculate the defect area in this diagram without using x in the calculation? 10. Using ideas similar to those in Questions 7 9, find an expression for the defect area that uses cos C and does not use y.
5 11. Earlier in your explorations, did you notice that the two defects area always had the same area? Use your expressions for the two defects area to explain why this is true. Putting It All Together the Law of Cosines! 12. Using the ideas you have discovered, write an equation that combines your observations about area into one sentence. (Assume the triangle is acute.) The area of Square III equals. 13. Using a 2, b 2, and c 2 for the areas of the squares and the expressions you found in Questions 9 and 10 for the areas of the defects, rewrite your area relationship into a formula (again, assume the triangle is acute). Check your formula with a classmate.
Trigonometry Learning Strategies. What should students be able to do within this interactive?
Trigonometry Learning Strategies What should students be able to do within this interactive? Identify a right triangle. Identify the acute reference angles. Recognize and name the sides, and hypotenuse
More informationA. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.
Algebra - Problem Drill 19: Basic Trigonometry - Right Triangle No. 1 of 10 1. Which of the following points lies on the unit circle? (A) 1, 1 (B) 1, (C) (D) (E), 3, 3, For a point to lie on the unit circle,
More information15 x. Substitute. Multiply. Add. Find the positive square root.
hapter Review.1 The Pythagorean Theorem (pp. 3 70) Dynamic Solutions available at igideasmath.com Find the value of. Then tell whether the side lengths form a Pythagorean triple. c 2 = a 2 + b 2 Pythagorean
More informationMath 144 Activity #7 Trigonometric Identities
144 p 1 Math 144 Activity #7 Trigonometric Identities What is a trigonometric identity? Trigonometric identities are equalities that involve trigonometric functions that are true for every single value
More informationMath 2 Trigonometry. People often use the acronym SOHCAHTOA to help remember which is which. In the triangle below: = 15
Math 2 Trigonometry 1 RATIOS OF SIDES OF A RIGHT TRIANGLE Trigonometry is all about the relationships of sides of right triangles. In order to organize these relationships, each side is named in relation
More informationAssumption High School BELL WORK. Academic institution promoting High expectations resulting in Successful students
BELL WORK Geometry 2016 2017 Day 51 Topic: Chapter 8.3 8.4 Chapter 8 Big Ideas Measurement Some attributes of geometric figures, such as length, area, volume, and angle measure, are measurable. Units are
More informationName Score Period Date. m = 2. Find the geometric mean of the two numbers. Copy and complete the statement.
Chapter 6 Review Geometry Name Score Period Date Solve the proportion. 3 5 1. = m 1 3m 4 m = 2. 12 n = n 3 n = Find the geometric mean of the two numbers. Copy and complete the statement. 7 x 7? 3. 12
More information8-2 Trigonometric Ratios
8-2 Trigonometric Ratios Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Write each fraction as a decimal rounded to the nearest hundredth. 1. 2. 0.67 0.29 Solve each equation. 3. 4. x = 7.25
More informationChapter 4 Trigonometric Functions
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios
More informationName Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean. A. Definitions: 1.
Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean A. Definitions: 1. Geometric Mean: 2. Right Triangle Altitude Similarity Theorem: If the altitude is
More informationGeometry Warm Up Right Triangles Day 8 Date
Geometry Warm Up Right Triangles Day 8 Name Date Questions 1 4: Use the following diagram. Round decimals to the nearest tenth. P r q Q p R 1. If PR = 12 and m R = 19, find p. 2. If m P = 58 and r = 5,
More informationAnswer Key. 7.1 Tangent Ratio. Chapter 7 Trigonometry. CK-12 Geometry Honors Concepts 1. Answers
7.1 Tangent Ratio 1. Right triangles with 40 angles have two pairs of congruent angles and therefore are similar. This means that the ratio of the opposite leg to adjacent leg is constant for all 40 right
More informationFind the geometric mean between 9 and 13. Find the geometric mean between
Five-Minute Check (over Lesson 8 1) CCSS Then/Now New Vocabulary Theorem 8.4: Pythagorean Theorem Proof: Pythagorean Theorem Example 1: Find Missing Measures Using the Pythagorean Theorem Key Concept:
More information5-7 The Pythagorean Theorem
5-7 The Pythagorean Theorem Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Classify each triangle by its angle measures. 1. 2. acute right 3. Simplify 12 4. If a = 6, b = 7, and c = 12, find
More informationAssignment 1 and 2: Complete practice worksheet: Simplifying Radicals and check your answers
Geometry 0-03 Summary Notes Right Triangles and Trigonometry These notes are intended to be a guide and a help as you work through Chapter 8. These are not the only thing you need to read, however. Rely
More informationExercise Solutions for Introduction to 3D Game Programming with DirectX 10
Exercise Solutions for Introduction to 3D Game Programming with DirectX 10 Frank Luna, September 6, 009 Solutions to Part I Chapter 1 1. Let u = 1, and v = 3, 4. Perform the following computations and
More informationCHAPTER 5: Analytic Trigonometry
) (Answers for Chapter 5: Analytic Trigonometry) A.5. CHAPTER 5: Analytic Trigonometry SECTION 5.: FUNDAMENTAL TRIGONOMETRIC IDENTITIES Left Side Right Side Type of Identity (ID) csc( x) sin x Reciprocal
More informationThe Pythagorean Theorem & Special Right Triangles
Theorem 7.1 Chapter 7: Right Triangles & Trigonometry Sections 1 4 Name Geometry Notes The Pythagorean Theorem & Special Right Triangles We are all familiar with the Pythagorean Theorem and now we ve explored
More informationCK- 12 Algebra II with Trigonometry Concepts 1
1.1 Pythagorean Theorem and its Converse 1. 194. 6. 5 4. c = 10 5. 4 10 6. 6 5 7. Yes 8. No 9. No 10. Yes 11. No 1. No 1 1 1. ( b+ a)( a+ b) ( a + ab+ b ) 1 1 1 14. ab + c ( ab + c ) 15. Students must
More informationThe graph of a proportional relation always contains the origin and has a slope equal to the constant of proportionality.
Chapter 11.1 Ratios and Rates A ratio is a comparison of two numbers, a and b, by division. The numbers a and b are called terms of the ratio. A ratio can be expressed in three different ways. 1. Word
More information2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.
1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving
More informationBrunswick School Department Honors Geometry Unit 6: Right Triangles and Trigonometry
Understandings Questions Knowledge Vocabulary Skills Right triangles have many real-world applications. What is a right triangle? How to find the geometric mean of two numbers? What is the Pythagorean
More information8.6 Inverse Trigonometric Ratios
www.ck12.org Chapter 8. Right Triangle Trigonometry 8.6 Inverse Trigonometric Ratios Learning Objectives Use the inverse trigonometric ratios to find an angle in a right triangle. Solve a right triangle.
More informationTImath.com Algebra 1. Trigonometric Ratios
Algebra 1 Trigonometric Ratios ID: 10276 Time required 60 minutes Activity Overview In this activity, students discover the trigonometric ratios through measuring the side lengths of similar triangles
More informationUnit 4-Review. Part 1- Triangle Theorems and Rules
Unit 4-Review - Triangle Theorems and Rules Name of Theorem or relationship In words/ Symbols Diagrams/ Hints/ Techniques 1. Side angle relationship 2. Triangle inequality Theorem 3. Pythagorean Theorem
More information7.1 Right Triangle Trigonometry; Applications Objectives
Objectives 1. Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles. Use the Complimentary Angle Theorem 3. Solve Right Triangles 4. Solve Applied Problems 9 November 017 1 Kidoguchi,
More informationMIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 FALL 2018 KUNIYUKI 150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART
Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 FALL 018 KUNIYUKI 150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART Show all work, simplify as appropriate, and
More informationAlgebra 1B. Unit 9. Algebraic Roots and Radicals. Student Reading Guide. and. Practice Problems
Name: Date: Period: Algebra 1B Unit 9 Algebraic Roots and Radicals Student Reading Guide and Practice Problems Contents Page Number Lesson 1: Simplifying Non-Perfect Square Radicands 2 Lesson 2: Radical
More informationName: for students entering. Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2
Name: Richard Montgomery High School Department of Mathematics Summer Math Packet for students entering Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2 (Please go the RM website
More informationGeometry Note Cards EXAMPLE:
Geometry Note Cards EXAMPLE: Lined Side Word and Explanation Blank Side Picture with Statements Sections 12-4 through 12-5 1) Theorem 12-3 (p. 790) 2) Theorem 12-14 (p. 790) 3) Theorem 12-15 (p. 793) 4)
More information8-2 The Pythagorean Theorem and Its Converse. Find x. 27. SOLUTION: The triangle with the side lengths 9, 12, and x form a right triangle.
Find x. 27. The triangle with the side lengths 9, 12, and x form a right triangle. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
More information8 Right Triangle Trigonometry
www.ck12.org CHAPTER 8 Right Triangle Trigonometry Chapter Outline 8.1 THE PYTHAGOREAN THEOREM 8.2 CONVERSE OF THE PYTHAGOREAN THEOREM 8.3 USING SIMILAR RIGHT TRIANGLES 8.4 SPECIAL RIGHT TRIANGLES 8.5
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.1 Trigonometric Identities Copyright Cengage Learning. All rights reserved. Objectives Simplifying Trigonometric Expressions Proving
More informationPre-AP Geometry 8-2 Study Guide: Trigonometric Ratios (pp ) Page! 1 of! 14
Pre-AP Geometry 8-2 Study Guide: Trigonometric Ratios (pp 541-544) Page! 1 of! 14 Attendance Problems. Write each fraction as a decimal rounded to the nearest hundredths. 2 7 1.! 2.! 3 24 Solve each equation.
More informationTrigonometric Functions. Copyright Cengage Learning. All rights reserved.
4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.3 Right Triangle Trigonometry Copyright Cengage Learning. All rights reserved. What You Should Learn Evaluate trigonometric
More informationName, Date, Period. R Θ R x R y
Name, Date, Period Virtual Lab Vectors & Vector Operations Setup 1. Make sure your calculator is set to degrees and not radians. Sign out a laptop and power cord. Plug in the laptop and leave it plugged
More informationDuring: The Pythagorean Theorem and Its converse
Before: November 1st As a warm-up, let's do the Challenge Problems from the 5.1-5.4 Quiz Yesterday 1. In Triangle ABC, centroid D is on median AM. AD = x - 3 and DM = 3x - 6. Find AM. 2. In Triangle ABC,
More informationPrimary Trigonometric Ratios
What s important in this lesson: Suggested time: 75 minutes Primary Trigonometric Ratios MFM2P1 Unit4 Lesson3 StudentinstructionSheet Questions for the teacher: 2. Assessment and Evaluation Sheet 1. The
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationMath 521B Trigonometry Assignment
Math 521B Trigonometry Assignment Multiple Choice Identify the choice that best completes the statement or answers the question. 1. What is the reference angle for 200 in standard position? A 100 C 20
More informationSOH CAH TOA. b c. sin opp. hyp. cos adj. hyp a c. tan opp. adj b a
SOH CAH TOA sin opp hyp b c c 2 a 2 b 2 cos adj hyp a c tan opp adj b a Trigonometry Review We will be focusing on triangles What is a right triangle? A triangle with a 90º angle What is a hypotenuse?
More informationBasic Trigonometry. Trigonometry deals with the relations between the sides and angles of triangles.
Basic Trigonometry Trigonometry deals with the relations between the sides and angles of triangles. A triangle has three sides and three angles. Depending on the size of the angles, triangles can be: -
More information1 The six trigonometric functions
Spring 017 Nikos Apostolakis 1 The six trigonometric functions Given a right triangle, once we select one of its acute angles, we can describe the sides as O (opposite of ), A (adjacent to ), and H ().
More information5.5 Special Rights. A Solidify Understanding Task
SECONDARY MATH III // MODULE 5 MODELING WITH GEOMETRY 5.5 In previous courses you have studied the Pythagorean theorem and right triangle trigonometry. Both of these mathematical tools are useful when
More informationMATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions
Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles
More informationAs we know, the three basic trigonometric functions are as follows: Figure 1
Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an
More informationPart 1: Using Vectors to Approximate the Neutral Current in a Three Phase Power System
Part 1: Using Vectors to Approximate the Neutral Current in a Three Phase Power System for three phase 60 hertz systems with unity power factor and resistive loads by Gerald Newton November 3, 1999 The
More informationDirections: Examine the Unit Circle on the Cartesian Plane (Unit Circle: Circle centered at the origin whose radius is of length 1)
Name: Period: Discovering the Unit Circle Activity Secondary III For this activity, you will be investigating the Unit Circle. You will examine the degree and radian measures of angles. Note: 180 radians.
More informationPART 1: USING SCIENTIFIC CALCULATORS (50 PTS.)
Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 SPRING 2018 KUNIYUKI 150 POINTS TOTAL: 50 FOR PART 1, AND 100 FOR PART 2 Show all work, simplify as appropriate,
More informationName: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions
Name: Math Analysis Chapter 3 Notes: Eponential and Logarithmic Functions Day : Section 3-1 Eponential Functions 3-1: Eponential Functions After completing section 3-1 you should be able to do the following:
More informationSail into Summer with Math!
Sail into Summer with Math! For Students Entering Math C This summer math booklet was developed to provide students an opportunity to review grade level math objectives and to improve math performance.
More informationGeometry Unit 7 - Notes Right Triangles and Trigonometry
Geometry Unit 7 - Notes Right Triangles and Trigonometry Review terms: 1) right angle ) right triangle 3) adjacent 4) Triangle Inequality Theorem Review topic: Geometric mean a = = d a d Syllabus Objective:
More informationVectors in Component Form
Vectors in Component Form Student Activity - Answers 7 8 9 10 11 12 Introduction TI-Nspire & TI-Nspire CAS Investigation Student 30 min Vectors are used to represent quantities that have both magnitude
More informationUnit 3 Practice Test Questions Trigonometry
Unit 3 Practice Test Questions Trigonometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. How you would determine the indicated angle measure, if it
More information(+4) = (+8) =0 (+3) + (-3) = (0) , = +3 (+4) + (-1) = (+3)
Lesson 1 Vectors 1-1 Vectors have two components: direction and magnitude. They are shown graphically as arrows. Motions in one dimension form of one-dimensional (along a line) give their direction in
More informationChapter. Triangles. Copyright Cengage Learning. All rights reserved.
Chapter 3 Triangles Copyright Cengage Learning. All rights reserved. 3.5 Inequalities in a Triangle Copyright Cengage Learning. All rights reserved. Inequalities in a Triangle Important inequality relationships
More informationExercise Set 6.2: Double-Angle and Half-Angle Formulas
Exercise Set : Double-Angle and Half-Angle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin
More information: SINE, COSINE, & TANGENT RATIOS
Geometry Notes Packet Name: 9.2 9.4: SINE, COSINE, & TANGENT RATIOS Trigonometric Ratios A ratio of the lengths of two sides of a right triangle. For any acute angle, there is a leg Opposite the angle
More informationVector components and motion
Vector components and motion Objectives Distinguish between vectors and scalars and give examples of each. Use vector diagrams to interpret the relationships among vector quantities such as force and acceleration.
More informationWarm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72
Warm Up 1. What is the third angle measure in a triangle with angles measuring 65 and 43? 72 Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest degree.
More informationJUST THE MATHS SLIDES NUMBER 3.1. TRIGONOMETRY 1 (Angles & trigonometric functions) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 3.1 TRIGONOMETRY 1 (Angles & trigonometric functions) by A.J.Hobson 3.1.1 Introduction 3.1.2 Angular measure 3.1.3 Trigonometric functions UNIT 3.1 - TRIGONOMETRY 1 - ANGLES
More informationUnit 2 Review. Short Answer 1. Find the value of x. Express your answer in simplest radical form.
Unit 2 Review Short nswer 1. Find the value of x. Express your answer in simplest radical form. 30º x 3 24 y 6 60º x 2. The size of a TV screen is given by the length of its diagonal. The screen aspect
More information5.7 Justifying the Laws
SECONDARY MATH III // MODULE 5 The Pythagorean theorem makes a claim about the relationship between the areas of the three squares drawn on the sides of a right triangle: the sum of the area of the squares
More informationand sinθ = cosb =, and we know a and b are acute angles, find cos( a+ b) Trigonometry Topics Accuplacer Review revised July 2016 sin.
Trigonometry Topics Accuplacer Revie revised July 0 You ill not be alloed to use a calculator on the Accuplacer Trigonometry test For more information, see the JCCC Testing Services ebsite at http://jcccedu/testing/
More informationRight 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 informationJan 1 4:08 PM. We write this in a shorter manner for simplicity. leg
Review Pythagorean Theorem Jan 1 4:08 PM We write this in a shorter manner for simplicity. leg hyp leg or a c b Note, the last statement can be misleading if the letters used are not in the correct position.
More informationKey Concept Trigonometric Ratios. length of leg opposite A length of hypotenuse. = a c. length of leg adjacent to A length of hypotenuse
8-3 Trigonometry ommon ore State Standards G-SRT..8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. lso, G-SRT..7, G-MG..1 MP 1, MP 3, MP 4, MP Objective
More informationSection 8.2 Vector Angles
Section 8.2 Vector Angles INTRODUCTION Recall that a vector has these two properties: 1. It has a certain length, called magnitude 2. It has a direction, indicated by an arrow at one end. In this section
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationAlgebra and Trig. I. P=(x,y) 1 1. x x
Algebra and Trig. I 4.3 Right Angle Trigonometry y P=(x,y) y P=(x,y) 1 1 y x x x We construct a right triangle by dropping a line segment from point P perpendicular to the x-axis. So now we can view as
More informationsin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
More informationStudent Exploration: Vectors
Name: Date: Student Exploration: Vectors Vocabulary: component, dot product, magnitude, resultant, scalar, unit vector notation, vector Prior Knowledge Question (Do this BEFORE using the Gizmo.) An airplane
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationDetermining a Triangle
Determining a Triangle 1 Constraints What data do we need to determine a triangle? There are two basic facts that constrain the data: 1. The triangle inequality: The sum of the length of two sides is greater
More information7.4. The Primary Trigonometric Ratios. LEARN ABOUT the Math. Connecting an angle to the ratios of the sides in a right triangle. Tip.
The Primary Trigonometric Ratios GOL Determine the values of the sine, cosine, and tangent ratios for a specific acute angle in a right triangle. LERN OUT the Math Nadia wants to know the slope of a ski
More informationIntegrated Math II. IM2.1.2 Interpret given situations as functions in graphs, formulas, and words.
Standard 1: Algebra and Functions Students graph linear inequalities in two variables and quadratics. They model data with linear equations. IM2.1.1 Graph a linear inequality in two variables. IM2.1.2
More informationUsing this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
More informationIntegration by Triangle Substitutions
Integration by Triangle Substitutions The Area of a Circle So far we have used the technique of u-substitution (ie, reversing the chain rule) and integration by parts (reversing the product rule) to etend
More informationInstructions. Do not open your test until instructed to do so!
st Annual King s College Math Competition King s College welcomes you to this year s mathematics competition and to our campus. We wish you success in this competition and in your future studies. Instructions
More informationLT 2.1 Study Guide and Intervention Classifying Triangles
LT 2.1 Study Guide and Intervention Classifying Triangles Classify Triangles by Angles One way to classify a triangle is by the measures of its angles. If all three of the angles of a triangle are acute
More information2. Pythagorean Theorem:
Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle
More informationBig Ideas: determine an approximate value of a radical expression using a variety of methods. REVIEW Radicals
Big Ideas: determine an approximate value of a radical expression using a variety of methods. REVIEW N.RN. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
More information4.5 Linearization Calculus 4.5 LINEARIZATION. Notecards from Section 4.5: Linearization; Differentials. Linearization
4.5 Linearization Calculus 4.5 LINEARIZATION Notecards from Section 4.5: Linearization; Differentials Linearization The goal of linearization is to approximate a curve with a line. Why? Because it s easier
More informationTrigonometry.notebook. March 16, Trigonometry. hypotenuse opposite. Recall: adjacent
Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)
More informationActivity 8b - Electric Field Exploration
Name Date Activity 8b - Electric Field Exploration Pd Go to the following website: http://phet.colorado.edu Find the heading Run our Simulations and click On Line. Under the Simulations heading, select
More informationNorth Carolina Math 2 Transition Edition Unit 5 Assessment: Trigonometry
Name: Class: _ Date: _ North Carolina Math 2 Transition Edition Unit 5 Assessment: Trigonometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the
More informationUnit 2 - The Trigonometric Functions - Classwork
Unit 2 - The Trigonometric Functions - Classwork Given a right triangle with one of the angles named ", and the sides of the triangle relative to " named opposite, adjacent, and hypotenuse (picture on
More informationChapter 8 Test Wednesday 3/28
Chapter 8 Test Wednesday 3/28 Warmup Pg. 487 #1-4 in the Geo book 5 minutes to finish 1 x = 4.648 x = 40.970 x = 6149.090 x = -5 What are we learning today? Pythagoras The Rule of Pythagoras Using Pythagoras
More informationUniversity School of Nashville. Sixth Grade Math. Self-Guided Challenge Curriculum. Unit 4. Plane Geometry*
University School of Nashville Sixth Grade Math Self-Guided Challenge Curriculum Unit 4 Plane Geometry* This curriculum was written by Joel Bezaire for use at the University School of Nashville, funded
More informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
More informationVectors in Component Form
Vectors in Component Form Student Activity 7 8 9 10 11 12 Introduction TI-Nspire & TI-Nspire CAS Investigation Student 30 min Vectors are used to represent quantities that have both magnitude and direction.
More information6.1 George W. Ferris Day Off
6.1 George W. Ferris Day Off A Develop Understanding Task Perhaps you have enjoyed riding on a Ferris wheel at an amusement park. The Ferris wheel was invented by George Washington Ferris for the 1893
More informationPrecalculus Prerequisites Ridgewood High School
Precalculus Prerequisites Ridgewood High School 2013-2014 Future Precalculus Student: Mathematics builds! To be successful in Precalculus, there are certain skills that you are expected to have already
More informationTriangles and Vectors
Chapter 3 Triangles and Vectors As was stated at the start of Chapter 1, trigonometry had its origins in the study of triangles. In fact, the word trigonometry comes from the Greek words for triangle measurement.
More informationExample: x 10-2 = ( since 10 2 = 100 and [ 10 2 ] -1 = 1 which 100 means divided by 100)
Scientific Notation When we use 10 as a factor 2 times, the product is 100. 10 2 = 10 x 10 = 100 second power of 10 When we use 10 as a factor 3 times, the product is 1000. 10 3 = 10 x 10 x 10 = 1000 third
More informationG.1.f.: I can evaluate expressions and solve equations containing nth roots or rational exponents. IMPORTANT VOCABULARY. Pythagorean Theorem
Pre-AP Geometry Standards/Goals: C.1.f.: I can prove that two right triangles are congruent by applying the LA, LL, HL, and HA congruence statements. o I can prove right triangles are similar to one another.
More informationChapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas
More informationAnswer Explanations for: ACT June 2012, Form 70C
Answer Explanations for: ACT June 2012, Form 70C Mathematics 1) C) A mean is a regular average and can be found using the following formula: (average of set) = (sum of items in set)/(number of items in
More informationUnit 5, Lesson 4.3 Proving the Pythagorean Theorem using Similarity
Unit 5, Lesson 4.3 Proving the Pythagorean Theorem using Similarity Geometry includes many definitions and statements. Once a statement has been shown to be true, it is called a theorem. Theorems, like
More informationMath 8 Winter 2010 Midterm 2 Review Problems Solutions - 1. xcos 6xdx = 4. = x2 4
Math 8 Winter 21 Midterm 2 Review Problems Solutions - 1 1 Evaluate xcos 2 3x Solution: First rewrite cos 2 3x using the half-angle formula: ( ) 1 + cos 6x xcos 2 3x = x = 1 x + 1 xcos 6x. 2 2 2 Now use
More information