Page. Trigonometry Sine Law and Cosine Law. push
|
|
- Veronica Norton
- 7 years ago
- Views:
Transcription
1 Trigonometry Sine Law and Cosine Law Page Trigonometry can be used to calculate the side lengths and angle measures of triangles. Triangular shapes are used in construction to create rigid structures. Quadrilaterals, on the other hand, are not rigid structures. In the diagram below, you can see how a rectangle can be pushed to form a parallelogram. All sides remain the same length, but the interior angles change. push A rectangle, therefore, is not a rigid structure. Similarly, any polygon with four or more sides is not a rigid structure. As stated above, however, a triangle is a rigid structure. If you push a triangle, the shape c_annot change without changing the side lengths. push Due to the rigid structure of triangles, they are a common shape in construction of roof supports and tall towers. In the previous lesson, you studied different types of triangles, which included calculating the side lengths and angle measurements of right-angled triangles. In this lesson, you will learn how to find the lengths of sides and measurements of angles of triangles that are not right-angled triangles.
2 Trigonometry Sine Law and Cosine Law Page 2 Non-right triangles are triangles that do not have a 90 angle. The trigonometric ratios of sin, cos, and tan are used to find the unknown measurements in right triangles only To solve for the unknown measurements in triangles that do not have a right angle, you need different formulas. One of the foil tulas you can use is the Sine Law, also known as the Law of Sines. It may be helpful for you to add this formula to your resource sheet. You should also include the situations in which the Sine Law can be used. The Sine Law in words is: "The ratio of the sine of any angle over its opposite side is a constant." The Sine Law is written as follows: sin ZA sin ZB sin ZC a O r a sin LA sin LB - sin ZC As long as you know both an angle and its opposite side, the Sine Law can help you. The Sine Law can be used in two situations. Situation One Given two angles of a triangle and one of its sides, you can solve a non-right triangle by using these steps: 1. Find the third angle by subtracting the sum of the two given angles from Find the two missing sides using the Sine Law. You may remember that solving a triangle means finding the lengths of ALL the missing sides, and the measurements of ALL the missing angles. -2-
3 Trigonometry Sine Law and Cosine Law Page 3 Example 1 Find the lengths of side a. A t't Example 2 Solve the triangle by finding the lengths of sides a and b, and the measurement of LB. A -3-
4 Trigonometry Sine Law and Cosine Law Page 4 Situation Two If you are given the lengths of two sides and the measurement of one angle of a triangle, you can use the Sine Law, but only if the angle you are given is opposite one of the given sides. Example 3 Explain in which of the following triangles you could use the Sine Law. a) Example 4 Find the measurements of LA and LC of AABC, given the diagram below. -4-
5 Trigonometry Sine Law and Cosine Law Page 5 Example 5 Suppose you are the pilot of a commercial airliner. You find it necessary to detour around a group of thundershowers. You turn at an angle of 21 to your original path, fly for a while, turn, and intercept your original path at an angle of 35, 70 kilometres from where you left it. How much further did you have to go because of the detour? r 947,11' 3" N.,7 Example 6 70 Two observers on boats have located a sunken ship using sonar equipment. The sunken ship is in line between the two observers. The angle of depression from Observer A to the sunken ship is 40, and the angle of depression from Observer B to the sunken ship is 36. The distance between the observers is 450 metres. How far is the ship below the surface of the water? Round your answer to the nearest metre. B (water surface) N;P /01 S (sunken ship) ve Note that the depth is the perpendicular distance from the sunken ship to the surface of the water. 1. LA, LB, and LC are all angles in an isosceles triangle. If LA = 40 and LB and LC are base angles in the triangle, what is the measure of LC? 2. Which polygon has all equal sides and all interior angles equal to 90? 3. LA and LB are opposite interior angles in a parallelogram. If LA = 38, what is the measure of LB?
6 Trigonometry Sine Law and Cosine Law Page 6 1. Find the length of side b. a) b) 7-/ 2. Find the measurement of angle C. a) 3. Find the length of AR A * S if\ c -t -6-
7 Trigonometry Sine Law and Cosine Law Page 7 4. All and Beth are both looking at a hot air balloon. For Alf the angle of elevation is 40, and for Beth the angle of elevation is 58. How high is the balloon above ground if the distance between All and Beth is 600 metres? In the diagram, A represents All, B represents Beth, H represents the hot air balloon, and HC is the distance of the balloon above the ground. 1 Yu. k in 5. Fire ranger Jarnal is 20 km due west of fire ranger Raj. They spot a fire and report the sightings as follows: m Jamal reports that the fire is N 47 0 E of his station. m Raj reports that the fire is N 20 W of his station. How far is the fire from Jamal's station? > 8 Write the values of LAJR and LARJ into the diagram before you solve the problem. -7-
8 Trigonometry Sine Law and Cosine Law Page 8 The Cosine Law can be used to solve certain non-right triangles. just as the Sine Law could only be used in certain situations, it is also true that the Cosine Law can only be used in certain situations. Make sure you add this formula to your resource sheet, along with the situations in which you can use the Cosine Law. The Cosine Law is written as follows for any triangle, ABC: a2 = b2 + C2-2bc cos LA b2 (12 c2 _ 7a c cos GB c2 = a2 + b2-2ab cos GC Note that there axe three versions of this formula, but there is a pattern. Notice the side squared on the left of the formula always corresponds to the angle on the right side of the formula. As with the Sine Law, in order for the Cosine Law to work, you must label your triangles correctly, where a is opposite GA, b is opposite GB, and so on Just like the Sine Law, the Cosine Law can be used in only two situations. Situation One If you are given two sides and the angle between them you can solve the triangle using the following steps. 1. Calculate the length of the missing side using the Cosine Law. 2. If required, you can use the Sine Law to calculate the measure of the smallest of the unknown angles. The Sine Law may be used more accurately with smaller angles. 3. If required, you can find the third angle using the fact that the sum of angles in a triangle is 180'. -8-
9 Trigonometry Sine Law and Cosine Law Page 9 Example 1 Given the triangle measurements as shown, find the measure of side a. Example 2 Find the length of side :v. Situation Two If you are given three sides of a non-right triangle, you can solve the triangle using the following steps: 1. Determine the measure of one of the larger angles using the Cosine Law (larger angles are opposite longer sides). 2, Calculate the measure of the smallest angle using the Sine Law. 3. Find the third angle using the fact that the sum of angles in a triangle is 180. L, _ ct l H kal1/41( -9-
10 Trigonometry -Sine Law and Cosine Law t it Page 10 USC Example 1 Find the measure of the smallest angle in triangle ABC, if a = 7, b = 8, and c =5. C. 7 :26 2- /1-, * Example 2 Solve for angles LA, LB, and LC in the following diagram. 2- t-, ( a - '
11 Trigonometry Sine Law and Cosine Law Page 11 Which formula should you use to solve the following problems the Sine Law or the Cosine Law? Do not solve the problems. a) Find the length of zu. LI) Find the length of t. c) Find the measure of LA. d) Find the measure of /D
12 Trigonometry Sine Law and Cosine Law Page Find the measure of the side or angle indicated. a) Find side b. b) Find angle A. A c) Find side a. Jot-. AB t 1 (,( C The arms of a "jaws" machine used for prying open crushed vehicles in accidents are 75 cm long. The jaws are joined at one end and open at various angles. If the angle formed by the arms is 55.6, how far apart are the tips? -12-
13 Trigonometry Sine Law and Cosine Law Page 13 The posts of a hockey goal are 2.0 m apart. A player attempts to score by shooting the puck along the ice from a point 6.5 m from one post and 8.0 m from the other post. Within what angle, 9, must the shot be made? (Give your answer to the nearest degree.) -13-
14
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 informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More information(15.) To find the distance from point A to point B across. a river, a base line AC is extablished. AC is 495 meters
(15.) To find the distance from point A to point B across a river, a base line AC is extablished. AC is 495 meters long. Angles
More informationThe Primary Trigonometric Ratios Word Problems
The Primary Trigonometric Ratios Word Problems. etermining the measures of the sides and angles of right triangles using the primary ratios When we want to measure the height of an inaccessible object
More informationHow To Solve The Pythagorean Triangle
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 informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationLaw 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 informationMathematics (Project Maths Phase 1)
2011. S133S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Ordinary Level Time: 2 hours 300 marks Running
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationCalculating Area, Perimeter and Volume
Calculating Area, Perimeter and Volume You will be given a formula table to complete your math assessment; however, we strongly recommend that you memorize the following formulae which will be used regularly
More information1. Introduction circular definition Remark 1 inverse trigonometric functions
1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationGeometry of 2D Shapes
Name: Geometry of 2D Shapes Answer these questions in your class workbook: 1. Give the definitions of each of the following shapes and draw an example of each one: a) equilateral triangle b) isosceles
More informationLesson 33: Example 1 (5 minutes)
Student Outcomes Students understand that the Law of Sines can be used to find missing side lengths in a triangle when you know the measures of the angles and one side length. Students understand that
More informationWednesday 15 January 2014 Morning Time: 2 hours
Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Wednesday 15 January 2014 Morning Time: 2 hours Candidate Number
More informationTrigonometry 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 informationGeometry - Calculating Area and Perimeter
Geometry - Calculating Area and Perimeter In order to complete any of mechanical trades assessments, you will need to memorize certain formulas. These are listed below: (The formulas for circle geometry
More information9. Trigonometry 2 - Sine, Cosine Rule, Area of 'Iriangle
9. Trigonometry 2 - Sine, Cosine Rule, Area of 'Iriangle Two yachts Ieave from harbour H. Yacht A sails on a bearing of 072o fbr 30 kilometres and stops. Yacht B sails on a bearin-e of 140' for 50 kilometres
More informationRight 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 informationSGS4.3 Stage 4 Space & Geometry Part A Activity 2-4
SGS4.3 Stage 4 Space & Geometry Part A Activity 2-4 Exploring triangles Resources required: Each pair students will need: 1 container (eg. a rectangular plastic takeaway container) 5 long pipe cleaners
More informationName: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given
More informationWEDNESDAY, 2 MAY 1.30 PM 2.25 PM. 3 Full credit will be given only where the solution contains appropriate working.
C 500/1/01 NATIONAL QUALIFICATIONS 01 WEDNESDAY, MAY 1.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper 1 (Non-calculator) 1 You may NOT use a calculator. Answer as many questions as you can. Full
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationIntermediate 2 NATIONAL QUALIFICATIONS. Mathematics Specimen Question Paper 1 (Units 1, 2, 3) Non-calculator Paper [C056/SQP105] Time: 45 minutes
[C056/SQP105] Intermediate Time: 45 minutes Mathematics Specimen Question Paper 1 (Units 1,, 3) Non-calculator Paper NATIONAL QUALIFICATIONS 1 Answer as many questions as you can. Full credit will be given
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical
More informationPythagoras Theorem. Page I can... 1... identify and label right-angled triangles. 2... explain Pythagoras Theorem. 4... calculate the hypotenuse
Pythagoras Theorem Page I can... 1... identify and label right-angled triangles 2... eplain Pythagoras Theorem 4... calculate the hypotenuse 5... calculate a shorter side 6... determine whether a triangle
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationTRIGONOMETRY Compound & Double angle formulae
TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae
More informationHow To Draw A Similar Figure From A Different Perspective
Chapter 6 Similarity of Figures 6.1 Similar Polygons 6.2 Determining if two Polygons are Similar 6.3 Drawing Similar Polygons 6.4 Similar Triangles 21 Name: 6.1 Similar Polygons A. What makes something
More information25 The Law of Cosines and Its Applications
Arkansas Tech University MATH 103: Trigonometry Dr Marcel B Finan 5 The Law of Cosines and Its Applications The Law of Sines is applicable when either two angles and a side are given or two sides and an
More informationWeek 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test
Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan
More information4.3 & 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 informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationGive 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 informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationSection 2.4 Law of Sines and Cosines
Section.4 Law of Sines and osines Oblique Triangle A triangle that is not a right triangle, either acute or obtuse. The measures of the three sides and the three angles of a triangle can be found if at
More informationIntroduction Assignment
PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationEstimating Angle Measures
1 Estimating Angle Measures Compare and estimate angle measures. You will need a protractor. 1. Estimate the size of each angle. a) c) You can estimate the size of an angle by comparing it to an angle
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationApplications for Triangles
Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationCumulative Test. 161 Holt Geometry. Name Date Class
Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2
More informationGeometry. Higher Mathematics Courses 69. Geometry
The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and
More informationMath 531, Exam 1 Information.
Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)
More informationIndirect Measurement Technique: Using Trigonometric Ratios Grade Nine
Ohio Standards Connections Measurement Benchmark D Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve
More informationAreas of Polygons. Goal. At-Home Help. 1. A hockey team chose this logo for their uniforms.
-NEM-WBAns-CH // : PM Page Areas of Polygons Estimate and measure the area of polygons.. A hockey team chose this logo for their uniforms. A grid is like an area ruler. Each full square on the grid has
More informationShape, Space and Measure
Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area
More informationGEOMETRIC MENSURATION
GEOMETRI MENSURTION Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the
More informationHS Mathematics Item Specification C1 TO
Task Model 1 Multiple Choice, single correct response G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of acute
More informationFORM 3 MATHEMATICS SCHEME C TIME: 30 minutes Non Calculator Paper INSTRUCTIONS TO CANDIDATES
DIRECTORATE FOR QUALITY AND STANDARDS IN EDUCATION Department for Curriculum Management and elearning Educational Assessment Unit Annual Examinations for Secondary Schools 2011 C FORM 3 MATHEMATICS SCHEME
More informationCalculus (6th edition) by James Stewart
Calculus (6th edition) by James Stewart Section 3.8- Related Rates 9. If and find when and Differentiate both sides with respect to. Remember that, and similarly and So we get Solve for The only thing
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More information/27 Intro to Geometry Review
/27 Intro to Geometry Review 1. An acute has a measure of. 2. A right has a measure of. 3. An obtuse has a measure of. 13. Two supplementary angles are in ratio 11:7. Find the measure of each. 14. In the
More informationUnit 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 informationCAMI Education linked to CAPS: Mathematics
- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to
More information9 Right Triangle Trigonometry
www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5
More informationUNIT H1 Angles and Symmetry Activities
UNIT H1 Angles and Symmetry Activities Activities H1.1 Lines of Symmetry H1.2 Rotational and Line Symmetry H1.3 Symmetry of Regular Polygons H1.4 Interior Angles in Polygons Notes and Solutions (1 page)
More informationGAP CLOSING. 2D Measurement GAP CLOSING. Intermeditate / Senior Facilitator s Guide. 2D Measurement
GAP CLOSING 2D Measurement GAP CLOSING 2D Measurement Intermeditate / Senior Facilitator s Guide 2-D Measurement Diagnostic...4 Administer the diagnostic...4 Using diagnostic results to personalize interventions...4
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More informationMake sure you get the grade you deserve!
How to Throw Away Marks in Maths GCSE One tragedy that only people who have marked eternal eamination papers such as GCSE will have any real idea about is the number of marks that candidates just throw
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationQuadrilaterals GETTING READY FOR INSTRUCTION
Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper
More information8-5 Angles of Elevation and Depression. The length of the base of the ramp is about 27.5 ft.
1.BIKING Lenora wants to build the bike ramp shown. Find the length of the base of the ramp. The length of the base of the ramp is about 27.5 ft. ANSWER: 27.5 ft 2.BASEBALL A fan is seated in the upper
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More informationUnit 8 Angles, 2D and 3D shapes, perimeter and area
Unit 8 Angles, 2D and 3D shapes, perimeter and area Five daily lessons Year 6 Spring term Recognise and estimate angles. Use a protractor to measure and draw acute and obtuse angles to Page 111 the nearest
More informationReal World Performance Tasks
Real World Performance Tasks Real World Real Life, Real Data, Real- Time - These activities put students into real life scenarios where they use real- time, real data to solve problems. In the Seriously
More informationMathematics Second Practice Test 1 Levels 4-6 Calculator not allowed
Mathematics Second Practice Test 1 Levels 4-6 Calculator not allowed Please read this page, but do not open your booklet until your teacher tells you to start. Write your name and the name of your school
More informationTarget To know the properties of a rectangle
Target To know the properties of a rectangle (1) A rectangle is a 3-D shape. (2) A rectangle is the same as an oblong. (3) A rectangle is a quadrilateral. (4) Rectangles have four equal sides. (5) Rectangles
More informationSample Test Questions
mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about
More informationChapter 3B - Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapter 3B - Vectors A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Vectors Surveyors use accurate measures of magnitudes and directions to
More informationPaper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Friday 10 June 2011 Morning Time: 1 hour 45 minutes
Centre No. Candidate No. Paper Reference 1 3 8 0 4 H Paper Reference(s) 1380/4H Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Higher Tier Friday 10 June 2011 Morning Time: 1 hour 45 minutes
More informationGeometry Unit 6 Areas and Perimeters
Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose
More informationTrigonometric 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 information11.3 Curves, Polygons and Symmetry
11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Edexcel IGCSE Centre Number Mathematics A Paper 3H Monday 6 June 2011 Afternoon Time: 2 hours Candidate Number Higher Tier Paper Reference 4MA0/3H You must have:
More information1. The volume of the object below is 186 cm 3. Calculate the Length of x. (a) 3.1 cm (b) 2.5 cm (c) 1.75 cm (d) 1.25 cm
Volume and Surface Area On the provincial exam students will need to use the formulas for volume and surface area of geometric solids to solve problems. These problems will not simply ask, Find the volume
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel International GCSE Mathematics A Paper 1FR Centre Number Tuesday 6 January 2015 Afternoon Time: 2 hours Candidate Number Foundation Tier Paper Reference
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel International GCSE Mathematics A Paper 3HR Centre Number Tuesday 6 January 015 Afternoon Time: hours Candidate Number Higher Tier Paper Reference
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationSection 2.3 Solving Right Triangle Trigonometry
Section.3 Solving Rigt Triangle Trigonometry Eample In te rigt triangle ABC, A = 40 and c = 1 cm. Find a, b, and B. sin 40 a a c 1 a 1sin 40 7.7cm cos 40 b c b 1 b 1cos40 9.cm A 40 1 b C B a B = 90 - A
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationAUTUMN UNIT 3. first half. Perimeter. Centimetres and millimetres. Metres and centimetres. Area. 3D shapes PART 3 MEASURES AND PROPERTIES OF SHAPES
PART AUTUMN first half MEASURES AND PROPERTIES OF SHAPES SECTION Perimeter SECTION Centimetres and millimetres SECTION Metres and centimetres SECTION Key Stage National Strategy CROWN COPYRIGHT 00 Area
More informationThe Triangle and its Properties
THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three
More informationMathematics. Steps to Success. and. Top Tips. Year 5
Pownall Green Primary School Mathematics and Year 5 1 Contents Page 1. Multiplication and Division 3 2. Positive and Negative Numbers 4 3. Decimal Notation 4. Reading Decimals 5 5. Fractions Linked to
More informationGeneral Certificate of Secondary Education January 2014. Mathematics Unit T3 (With calculator) Higher Tier [GMT31] FRIDAY 10 JANUARY, 9.15am 11.
Centre Number 71 Candidate Number General Certificate of Secondary Education January 2014 Mathematics Unit T3 (With calculator) Higher Tier [GMT31] MV18 FRIDAY 10 JANUARY, 9.15am 11.15 am TIME 2 hours,
More informationWednesday 6 November 2013 Morning
H Wednesday 6 November 2013 Morning GCSE MATHEMATICS B J567/03 Paper 3 (Higher Tier) *J540550313* Candidates answer on the Question Paper. OCR supplied materials: None Other materials required: Geometrical
More information