# CHAPTER 7 TRAVERSE Section I. SELECTION OF TRAVERSE DEFINITION

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 7 TRAVERSE Section I. SELECTION OF TRAVERSE DEFINITION A traverse is a series of straight lines called traverse legs. The surveyor uses them to connect a series of selected points called traverse stations (TS). The surveyor makes distance and angle measurements and uses them to compute the relative positions of the traverse stations on some system of coordinates. STARTING CONTROL Since the purpose of a traverse is to locate points relative to each other on a common grid, the surveyor needs certain elements of starting data, such as the coordinates of a starting point and an azimuth to an azimuth mark. There are several ways in which the starting data can be obtained, and the surveyor should make an effort to use the best data available to begin a traverse. The different variations in starting control are grouped into several general categories. Known Control Available Survey control is available in the form of existing stations with the station data published in a trigonometric list, or higher headquarters may establish the station and provide the station data. The surveyor obtains the azimuth to an azimuth mark (starting direction) by referring to a trigonometric list or computing from known coordinates. Maps Available When survey control is not available in the area, the surveyor must assume the coordinate of the starting station. The assumed data approximates the correct coordinate as closely as possible to facilitate operations. When a map of the area is available, the approximate coordinate of the starting station is scaled from the map. (For survey purposes, starting data scaled from a map are considered to be assumed data.) The surveyor determines the starting direction by scaling it from the map. No Maps Available When neither survey control nor maps are available, the coordinate of the starting point is assumed. The surveyor determines the starting direction by the most accurate means available. 7-1

2 TYPES OF TRAVERSE Construction surveying makes use of two basic types of traverse open traverse and closed traverse. Open Traverse An open traverse (figure 7-1) originates at a starting station, proceeds to its destination, and ends at a station whose relative position is not previously known. The open traverse is the least desirable type of traverse because it provides no check on fieldwork or starting data. For this reason, the planning of a traverse always provides for closure of the traverse. Traverses are closed in all cases where time permits. Closed Traverse A closed traverse starts at a point and ends at the same point or at a point whose relative position is known. The surveyor adjusts the measurements by computations to minimize the effect of accidental errors made in the measurements. Large errors are corrected. Traverse closed on starting point. A traverse which starts at a given point, proceeds to its destination, and returns to the starting point without crossing itself in the process is referred to as a loop traverse (figure 7-2). The surveyor uses this type of traverse to provide control of a tract or parcel boundary, and data for the area computation within the boundary. This type of traverse is also used if there is little or no existing control in the area and only the relative position of the points is required. A loop traverse starts and ends on a station of assumed coordinates and azimuth without affecting the computations, area, or relative position of the stations. If, however, the coordinates must be tied to an existing grid system, the traverse starts from a known station and azimuth on that system. While the loop traverse provides some check upon the fieldwork and computations, it does not provide for a check of starting data or insure detection of all the systematic errors that may occur in the survey. Traverse closed on second known point. A traverse closed on a second known point begins at a point of known coordinates, moves through the required point(s), and terminates at a second point of known coordinates. The surveyor prefers this type of traverse because it provides a check on the fieldwork, computations, and starting data. It also provides a basis for comparison to determine the overall accuracy of the work. 7-2

3 FIELDWORK In a traverse, three stations are considered to be of immediate significance. Surveyors refer to these stations as the rear station, the occupied station, and the forward station. The rear station is the station from which the surveyors performing the traverse have just moved or a point to which the azimuth is known. The occupied station is the station at which the party is located and over which the instrument is set. The immediate destination of the party is the forward station or the next station in succession. Horizontal Angles Always measure horizontal angles at the occupied station by sighting the instrument at the rear station and measuring the clockwise angles to the forward station. To Section II. FIELD SURVEY measure horizontal angles, make instrument pointings to the lowest visible point of the target which marks the rear and forward stations. Distance Measure the distance in a straight line between the occupied station and the forward station. Use horizontal taping procedures or electronic distance measuring equipment. TRAVERSE STATIONS The surveyor selects sites for traverse stations as the traverse progresses. The surveyor locates the stations in such a way that at any one station both the rear and forward stations are visible. 7-3

4 Selection of Stations If the distance is measured with tape, the line between stations must be free of obstacles for the taping team. The surveyor should keep the number of stations in a traverse to a minimum to reduce the accumulation of instrumental errors and the amount of computing required. Short traverse legs require the establishment and use of a greater number of stations and may cause excessive errors in azimuth because small errors in centering the instrument, in station marking equipment, and in instrument pointings are magnified and absorbed in the azimuth closure as errors in angle measurement. Station Markers Traverse station markers are usually 2- by 2-inch wooden stakes, 6 inches or more in length. The surveyor drives these stakes, called hubs, flush with the ground. The center of the top of the hub is marked with a surveyor s tack or with an X to designate the exact point of reference for angular and linear measurements. To assist in recovering the station, the surveyor drives a reference (witness) stake into the ground so that it slopes toward the station. The surveyor must write the identification of the station on the reference stake with a lumber crayon or china marking pencil or on a tag attached to the stake. Signal cloth may also be tied to the reference stake to further assist in identifying or recovering the station. Station Signals A signal must be erected over survey stations to provide a sighting point for the instrument operator and to serve as a reference for tape alignment by the taping team. The most commonly used signal is the range pole. When measuring angles, the surveyor places the tapered point of the range pole on the station mark and uses a rod level to make the pole vertical for observations. The surveyor should check the verticality of the pole by verifying that the bubble remains centered at other points on the range pole. The range pole is maintained in a vertical position throughout the observation period either by use of a range pole tripod or by someone holding the pole. To prevent the measurement of angles to the wrong point, the surveyor places the range pole in a vertical position only when it is being used to mark a survey station. ORGANIZATION OF TRAVERSE PARTY The number of personnel available to perform survey operations depends on the unit s Table of Organization and Equipment (TOE). The organization of these people into a traverse party and the duties assigned to each member will depend on the unit s Standing Operating Procedure (SOP). The organization and duties of traverse party members are based on the functional requirements of the traverse. Chief of the Party The chief of the party selects and marks the locations for the traverse stations and supervises the work of the other members of the party. The chief of the party also assists in the reconnaissance and planning of the survey. Instrument Operator The instrument operator measures the horizontal angles at each traverse station. Recorder The recorder keeps the field notes (see appendix B) for the party in a field notebook, and records the angles measured by the instrument operator, the distances measured by the tapeman, and all other data pertaining to the survey. The recorder is usually the party member designated to check the taped distances by pacing between traverse stations. Tapemen Two tapemen measure the distance from one traverse station to the next. 7-4

5 Rodman The rodman assists the chief of the party in marking the traverse stations, removes the target from the rear station when signaled by the instrument operator, and moves the target forward to the next traverse station. Variations The number of party members maybe reduced by combining the positions of party chief, instrument operator, and recorder. It is possible that one of the tapemen may double as a rodman, but it is best to have two tapemen and a separate rodman. It is possible to reduce the number of personnel in a traverse party, but there should be two tapemen and a separate rodman. Section III. COMPUTATIONS AZIMUTH COMPUTATION The azimuth of a line is defined as the horizontal angle, measured clockwise, from a base direction to the line in question. Depending upon the starting data and the desired results, the base direction used will be grid north. In extreme circumstances, the starting azimuth may even be a magnetic azimuth. In order for a traverse to be computed, the surveyor determines an azimuth for each leg of the traverse (figure 7-2). The surveyor must determine an azimuth for each leg of the traverse. The azimuth for each succeeding leg of the traverse is determined by adding the value of the measured angle at the occupied station to the azimuth from the occupied station to the rear station. The example which follows illustrates this procedure. It should be noted that on occupation of each successive station, the first step is to compute the back azimuth of the preceding traverse leg (the azimuth from the occupied station to the rear station). 7-5

6 Example Given: Azimuth from station A to azimuth mark (Az) Angle azimuth mark-a-b Angle A-B-C Angle B-C-A Angle C-A-azimuth mark Required: All leg azimuths Solution 7-6

7 AZIMUTH ADJUSTMENT The surveyor must determine the need for adjustment before beginning final coordinate computations. If the angular error of closure falls within computed allowable error, the surveyor may adjust the azimuths of the traverse. Allowable Angular Error (AE) The allowable angular error is determined by the formula AE = 1 or 20" per station. whichever is less, where N is the number of traverse stations. If azimuth error does not fall within allowable error, the surveyor must reobserve the station angles of the traverse in the field. In the example given, N equals 3. Therefore, 60 seconds is the allowable error. Azimuth Corrections Prior to determining a correction, the surveyor computes the actual error. The azimuth error is obtained by subtracting the computed closing azimuth from the known closing azimuth. This subtraction provides the angular error with the appropriate sign. By reversing this sign, the azimuth correction with the appropriate sign is obtained. For example, the azimuth from point A to an azimuth mark is The closing azimuth of a traverse to the same azimuth mark is determined to be This falls within allowable limits, so the surveyor may compute the error and correction as follows: Azimuth error Azimuth correction = computed azimuth - known azimuth = = =

9 7-9

10 COORDINATE COMPUTATIONS If the coordinate of a point and the azimuth and distance from that point to a second point are known, the surveyor can compute the coordinate of the second point. In figure 7-5, the coordinate of station A is known and the coordinate of B is to be determined. The azimuth and distance from station A to B are determined by measuring the horizontal angle from the azimuth mark to station B and the distance from station A to B. The grid casting and northing lines through both stations are shown. Since the grid is a rectangular system and the casting and northing lines form right angles at the point of intersection, the computation of the difference in northing (side dn) and difference in casting (side de) employs the formulas for the computation of a right triangle. The distance from A to B is the hypotenuse of the triangle, and the bearing angle (azimuth) is the known angle. The following formulas are used to compute dn and de: dn = Cos Azimuth x Distance de = Sin Azimuth x Distance In figure 7-5, the traverse leg falls in the first (northeast) quadrant since the value of the casting increases as the line goes east, and the value of the northing increases as it goes north. Both the de and dn are positive and are added to the casting and northing of station A to obtain the coordinate of station B. If the surveyor uses a calculator with trigonometric functions to compute the traverse, the azimuth is entered directly and the machine provides the correct sign of the function and the dn and de. If the functions are taken from tables, the computer provides the sign of the function by inspection. All lines going north have positive dns; south lines have negative. Lines going east have positive des; west lines have negative. Figure 7-6 illustrates the relationship of quadrant to sign. 7-10

11 DETERMINATION OF dn AND de In figure 7-7 (page 7-13), the azimuth from A-B is and the distance is dn = Cos X dn = X = de = Sin x de = X = The azimuth from B-C is and the distance is (note SE quadrant). dn = Cos X dn = X = de = Sin x de = X = The azimuth from C-A is , and the distance is (note NW quadrant). dn = Cos X dn = X = de = Sin x de = X = ACCURACY AND SPECIFICATIONS The overall accuracy of a traverse depends on the equipment and methods used in the measurements, the accuracy achieved, and the accuracy of the starting and closing data. An accuracy ratio of 1:5,000 is normally sought in construction surveying. In obtaining horizontal distances, an accuracy of at least 0.02 foot per 100 feet must be obtained. When using a one-minute instrument, the surveyor turns the horizontal angles once in each position, namely, one direct and one indirect, with angular closure of 20 seconds per station or 1 whichever is less. Linear Error To determine the acceptability of a traverse, the surveyor must compute the linear error of closure and the allowable error or the accuracy ratio or both. The first step in either case is to determine the linear error in dn and de. In the case of a loop traverse, the algebraic sum of the dns should equal zero. Any discrepancy is the linear error in dn. The same is true for des. Linear Error dn (en) = Linear Error de (ee) = The surveyor computes the linear error of closure (el) using the Pythagorean theorum. 7-11

12 Allowable Error The surveyor then computes the allowable error (AE) using the appropriate accuracy ratio (1/5,000 or 1/3,000) and the total length of the traverse. COORDINATE ADJUSTMENT The surveyor makes the adjustment of the traverse using the compass rule. The compass rule says that, for any leg of the traverse, the correction to be given to the dn or de is to the total correction for dn or de as the length of the leg is to the total length of the traverse. The total correction for dn or de is numerically equal to the en or ee, but with the opposite sign. Formulas In figure 7-7, the dn is and the de is and the total corrections are and +0.05, respectively. Note the following formulas: dn Correction per station = Total dn Correction x Distance to Station Length of Traverse Compare this to the linear error of closure. If the AE is greater than the el, the traverse is good and can be adjusted. If it is not good, it must be rerun. Ratio of Accuracy The ratio of accuracy provides a method of determining the traverse accuracy and comparing it to established standards. The ratio of accuracy is the ratio of the el to the total length of the traverse, after it is reduced to a common ratio and rounded down. For the first leg in the traverse in figure 7-7, the corrections are computed as follows: If the accuracy ratio does not fall within allowable limits, the traverse must be rerun. It is very possible that the distances as measured are correct and that the error can be attributed to large, compensating angular errors. Loop Traverse When adjusting a loop traverse, the surveyor applies the correction to the dns and des prior to computing the coordinates. The total correction must equal the total error. Sometimes, due to round off, the total 7-12

13 correction will not equal the total error. If this happens and the difference is one, reduce the correction to the shortest line or increase the longest line. When the error is greater than one, you may arbitrarily reduce/increase the corrections until the total correction equals the total error. The coordinate of the previous station ±dn± correction equals the coordinate of the next station. From figure 7-7, you compute as follows: When adjusting a traverse that starts and ends on two different stations, the surveyor computes the coordinates before the error is determined. In this case, the correction per leg is determined in the same manner as shown, but the correction is applied directly to the coordinates. The correction to be applied after the first leg is equal to the correction computed for the first leg. The correction to be applied after the second leg is equal to the correction for the first leg plus the correction computed for the second leg. The correction for the third leg equals the first correction plus the second correction plus the correction computed for the third leg and so on throughout the traverse. The last correction must be equal to the total correction required. 7-13

### CHAPTER 9 SURVEYING TERMS AND ABBREVIATIONS

CHAPTER 9 SURVEYING TERMS AND ABBREVIATIONS Surveying Terms 9-2 Standard Abbreviations 9-6 9-1 A) SURVEYING TERMS Accuracy - The degree of conformity with a standard, or the degree of perfection attained

### CURVES Section I. SIMPLE HORIZONTAL CURVES

CHAPTER 3 CURVES Section I. SIMPLE HORIZONTAL CURVES CURVE POINTS By studying TM 5-232, the surveyor learns to locate points using angles and distances. In construction surveying, the surveyor must often

### Algebra Geometry Glossary. 90 angle

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

### Geometry and Measurement

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

### 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

### 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

### Standard Surveying Terms

Standard Surveying Terms Aliquot - The description of fractional section ownership used in the U.S. public land states. A parcel is generally identified by its section, township, and range. The aliquot

### Parallel 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

Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation ED 5661 Mathematics & Navigation Teacher Institute August 2011 By Serena Gay Target: Precalculus (grades 11 or 12) Lesson

### 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

### CHAPTER 8 - LAND DESCRIPTIONS

CHAPTER 8 - LAND DESCRIPTIONS Notes: While the location of land is commonly referred to by street number and city, it is necessary to use the legal description in the preparation of those instruments relating

### 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.

### PHYSICS 151 Notes for Online Lecture #6

PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities

DUTIES OF THE CONSTRUCTION SURVEYOR In support of construction activities, the surveyor obtains the reconnaissance and preliminary data which are necessary at the planning stage. During the construction

### 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

### 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

### STATE CONTROL SURVEY SPECIFICATIONS FOR PRIMARY CONTROL SURVEYS. Now Obsolete

STATE CONTROL SURVEY SPECIFICATIONS FOR PRIMARY CONTROL SURVEYS Now Obsolete Caution: This document has been prepared by scanning the original Specifications for Primary Control Surveys - 1984 and using

### Pythagorean 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

### (1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

### SAULT COLLEGE OF APPLIED ARTS AND TECHNOLOGY SAULT STE. MARIE, ONTARIO COURSE OUTLINE. SURVEYING (Introduction to)

SAULT COLLEGE OF APPLIED ARTS AND TECHNOLOGY SAULT STE. MARIE, ONTARIO COURSE OUTLINE COURSE TITLE: SURVEYING (Introduction to) CODE NO. : SEMESTER: PROGRAM: AUTHOR: CIVIL/ARCHITECTURAL/CONSTRUCTION S.

### DEFINITIONS. 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

### 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

### 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

### Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º.

Using the Quadrant Eye Piece Protractor Handle You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements 90º. Plumb Bob ø

### 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

### INDEX. SR NO NAME OF THE PRACTICALS Page No. Measuring the bearing of traverse lines, calculation of included angles and check.

INDEX SR NO NAME OF THE PRACTICALS Page No 1 Measuring the bearing of traverse lines, calculation of included angles and check. 1 2 To study the essential parts of dumpy level & reduction of levels 3 To

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### 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

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### (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

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Elev. A = 200.05, Elev. B = 325.25, horizontal distance = 1625 feet a. 7.7 b..077 c. 12.98 d. 1.28

1. If the percent grade is 7.25 % and the horizontal length of grade is 1625 feet, what is the change in elevation? a. 117.81 feet b. 1178.13 feet c. 11.78 feet d. 1.17 feet 2. In surveying, the distance

### FORMULA FOR FINDING THE SQUARE FEET OF A RECTANGLE L x W = A

UNIT I REAL ESTATE MATH AREA MEASUREMENTS FORMULA FOR FINDING THE SQUARE FEET OF A RECTANGLE L x W = A Where: A = Area L = Length W = Width If the length = 30 and the width = 20 20 x 30 = 600 Sq. Feet

### Introduction 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

### Lesson 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

### Biggar 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

### Unit 6 Direction and angle

Unit 6 Direction and angle Three daily lessons Year 4 Spring term Unit Objectives Year 4 Recognise positions and directions: e.g. describe and find the Page 108 position of a point on a grid of squares

### 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

### http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

### Introduction and Mathematical Concepts

CHAPTER 1 Introduction and Mathematical Concepts PREVIEW In this chapter you will be introduced to the physical units most frequently encountered in physics. After completion of the chapter you will be

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Semester 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

### Trigonometry. An easy way to remember trigonometric properties is:

Trigonometry It is possible to solve many force and velocity problems by drawing vector diagrams. However, the degree of accuracy is dependent upon the exactness of the person doing the drawing and measuring.

### Survey Ties Guidelines

North Carolina Board of Examiners for Engineers and Surveyors Survey Ties Guidelines The North Carolina Board of Examiners for Engineers and Surveyors is providing this document to serve as an interpretative

### MINIMUM STANDARDS OF ACCURACY, CONTENT AND CERTIFICATION FOR SURVEYS AND MAPS ARTICLE I. TYPES OF SURVEYS

MINIMUM STANDARDS OF ACCURACY, CONTENT AND CERTIFICATION FOR SURVEYS AND MAPS ARTICLE I. TYPES OF SURVEYS Current with material published in Conn.L.J. through 5/13/08 Sec. 20-300b-1. General There are

### 25 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

### Section 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,

### Chapter 7: Land Descriptions

Chapter 7: Land Descriptions 7. Land Descriptions An * in the left margin indicates a change in the statute, rule or text since the last publication of the manual. I. Introduction While the location of

### Wind Direction Smart Sensor (S-WDA-M003)

(S-WDA-M003) The Wind Direction smart sensor is designed to work with HOBO Stations. The smart sensor has a plug-in modular connector that allows it to be added easily to a HOBO Station. All sensor parameters

### CHAPTER 4 LEGAL DESCRIPTION OF LAND DESCRIBING LAND METHODS OF DESCRIBING REAL ESTATE

r CHAPTER 4 LEGAL DESCRIPTION OF LAND DESCRIBING LAND A legal description is a detailed way of describing a parcel of land for documents such as deeds and mortgages that will be accepted in a court of

### TRIGONOMETRY FOR ANIMATION

TRIGONOMETRY FOR ANIMATION What is Trigonometry? Trigonometry is basically the study of triangles and the relationship of their sides and angles. For example, if you take any triangle and make one of the

### 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:

### Right 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

### Mathematics, Basic Math and Algebra

NONRESIDENT TRAINING COURSE Mathematics, Basic Math and Algebra NAVEDTRA 14139 DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited. PREFACE About this course: This is a self-study

### 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

### 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

### The Mathematics of Engineering Surveying (1)

The Mathematics of ngineering Surveying (1) Scenario s a new graduate you have gained employment as a graduate engineer working for a major contractor that employs 2000 staff and has an annual turnover

### ALGEBRA 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

### ROAD SURVEYING Section I. RECONNAISSANCE SURVEY PREPARATION AND SCOPE

CHAPTER 2 ROAD SURVEYING Section I. RECONNAISSANCE SURVEY PREPARATION AND SCOPE The reconnaissance survey is an extensive study of an entire area that might be used for a road or airfield. Its purpose

### NSPS SURVEY TECHNICIAN CERTIFICATION PROGRAM LEVEL III SAMPLE EXAMINATION QUESTIONS

NSPS SURVEY TECHNICIAN CERTIFICATION PROGRAM LEVEL III SAMPLE EXAMINATION QUESTIONS NATIONAL SOCIETY OF PROFESSIONAL SURVEYORS October 2007 This booklet has been prepared to provide an example of what

### EXPERIMENTAL ERROR AND DATA ANALYSIS

EXPERIMENTAL ERROR AND DATA ANALYSIS 1. INTRODUCTION: Laboratory experiments involve taking measurements of physical quantities. No measurement of any physical quantity is ever perfectly accurate, except

### 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

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

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

### Indirect 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

### QUICKCLOSE HP35S Surveying Programs - Version 1.1

QUICKCLOSE HP35S Surveying Programs - Version 1.1 USER REFERENCE GUIDE - for Evaluation Important Note: Program steps are only included in the version of User Reference Guides available from resellers.

### Sample 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

### Motion Graphs. It is said that a picture is worth a thousand words. The same can be said for a graph.

Motion Graphs It is said that a picture is worth a thousand words. The same can be said for a graph. Once you learn to read the graphs of the motion of objects, you can tell at a glance if the object in

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### Model Virginia Map Accuracy Standards Guideline

Commonwealth of Virginia Model Virginia Map Accuracy Standards Guideline Virginia Information Technologies Agency (VITA) Publication Version Control Publication Version Control: It is the user's responsibility

### FUNDAMENTALS OF LANDSCAPE TECHNOLOGY GSD Harvard University Graduate School of Design Department of Landscape Architecture Fall 2006

FUNDAMENTALS OF LANDSCAPE TECHNOLOGY GSD Harvard University Graduate School of Design Department of Landscape Architecture Fall 2006 6106/ M2 BASICS OF GRADING AND SURVEYING Laura Solano, Lecturer Name

### SWC 211 Soil and Water Conservation Engineering PLANE TABLE SURVEY

PLANE TABLE SURVEY General: In case of plane table survey, the measurements of survey lines of the traverse and their plotting to a suitable scale are done simultaneously on the field. Following are the

### Definitions, 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

### Geometry 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

### Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

### BASIC SURVEYING - THEORY AND PRACTICE

OREGON DEPARTMENT OF TRANSPORTATION GEOMETRONICS 200 Hawthorne Ave., B250 Salem, OR 97310 (503) 986-3103 Ron Singh, PLS Chief of Surveys (503) 986-3033 BASIC SURVEYING - THEORY AND PRACTICE David Artman,

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### PROBLEM 2.9. sin 75 sin 65. R = 665 lb. sin 75 sin 40

POBLEM 2.9 A telephone cable is clamped at A to the pole AB. Knowing that the tension in the right-hand portion of the cable is T 2 1000 lb, determine b trigonometr (a) the required tension T 1 in the

### 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

### STATE OF ALASKA DEPARTMENT OF NATURAL RESOURCES DIVISION OF MINING, LAND AND WATER. GENERAL SURVEY INSTRUCTIONS EASEMENTS Authority 11 AAC 53

STATE OF ALASKA DEPARTMENT OF NATURAL RESOURCES DIVISION OF MINING, LAND AND WATER GENERAL SURVEY INSTRUCTIONS EASEMENTS Authority 11 AAC 53 These instructions define the survey and platting criteria unique

### Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### 8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

### 2312 test 2 Fall 2010 Form B

2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### With the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given:

Page 1 In game development, there are a lot of situations where you need to use the trigonometric functions. The functions are used to calculate an angle of a triangle with one corner of 90 degrees. By

### Square Roots and the Pythagorean Theorem

4.8 Square Roots and the Pythagorean Theorem 4.8 OBJECTIVES 1. Find the square root of a perfect square 2. Use the Pythagorean theorem to find the length of a missing side of a right triangle 3. Approximate

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### Chapter 5: Trigonometric Functions of Angles

Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has

### CHAPTER 4 EARTHWORK. Section I. PLANNING OF EARTHWORK OPERATIONS

CHAPTER 4 EARTHWORK Section I. PLANNING OF EARTHWORK OPERATIONS IMPORTANCE In road, railroad, and airfield construction, the movement of large volumes of earth (earthwork) is one of the most important

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### RIGHT 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

### The 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

### Applications of the Pythagorean Theorem

9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem

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.

### The 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

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering