# Maths Pack. For the University Certificates in Astronomy and Cosmology

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 Powers of ten Astronomy makes use of very large and very small numbers. They are usually expressed as powers of. Ten to the power of two is (or. Ten to the power of three is (or. Below you will see some examples of different powers of ten. Complete the missing entries below and also make sure you can key all these numbers into your pocket calculator., 000, 000, Check that you can also work backwards, converting numbers that show all the digits, into numbers represented as powers of ten , ,000,000 Multiplying powers of Multiplying ordinary numbers such as is straightforward, but when powers of ten are involved we need to look carefully at how the powers are combined. Study the examples below and then fill in the blanks in the remaining examples. Take care with the minus sign.,000,000,000,000,000,000 + The previous method for multiplying two powers of ten may be extended to cover products of several powers of ten: Dividing by powers of ten + When we divide by a power of ten, it is the same as multiplying by the power of ten with the opposite sign. So in the following example we divide by a large number resulting in a small number: Conversely if we divide by a small number the result is a large number: This idea reappears in the following examples: University of Central Lancashire Version.0 - Page

3 Complete the following examples without using a calculator. Then check your results using a calculator. Summary of rules for combining powers of x x y y x + y x y x y x y xy ( For explanation of this last rule, see under Powers in general. Scientific notation Some numbers are not just simple powers of ten. Here are a few examples., 000, 000, 000, 00, Many of the number you see in your course will be expressed as the product of a number between and and a power of ten. This is often called scientific notation. Practise entering each of the numbers above into your scientific calculator. When you perform a calculation involving a mixture of numbers it is normal to express the result in scientific notation, especially if the result is very large or very small Powers in general It is possible to raise any number to the power of something. It does not need to be a power of ten as the next example shows. 0 Now let us replace the by another number say. We would read this last example out loud as two raised to the power six. When the power is large you may find it useful to resort to a calculator to evaluate the result. Now evaluate the following in the same way: Here is an example of how to calculate powers when the number in question is a power of. We have seen that a number raised to the power of means multiply that number by itself times. So: ( ( ( ( ( ( ( ( ( This is summarised by the third of the rules in the summary of Powers of ten. Evaluate the following: ( ( 000 University of Central Lancashire Version.0 - Page

4 Square roots Consider the following examples: What number when raised to the power of gives the number? The answer ( in this case is the square root of the number in question. This is an important concept and there is a special symbol,, for this. 0 Expressions involving the square root can occur in equations. See for example radius and area of a circle in Rearranging equations. Logarithms Some physical processes such as the eye s response are by their very nature logarithmic. So the equations defining magnitudes contain logarithms. Logarithms can arise elsewhere in the maths or they can just be used as a convenient way of displaying a large range of numbers on the same plot. For a number that is a simple power of ten, its logarithm (or log is just the power of ten. log(0 log( ( log 000 log( log ( ( log log log( In simple cases like this we can work out the result in our head. More complicated numbers require the use of a scientific calculator. Most calculators have two types of logs. (Usually the keys have labels ln and log. All of our examples use logs to the base of ten, which can be found by using the log key on your calculator. Now complete the following examples without using a calculator using a calculator. ( log 00 ( ( ( ( log log log log log( The logarithms of numbers that are not simple powers of ten can be found using a pocket calculator or (log tables! log (. 0.0 log log log log 0.0 log 0. ( ( ( 00 log (.. 0 Try the following using a calculator: log log( 0 log( 0. log( 000 If we need to find the logarithm of an expression, this is usually written in brackets, and we must work this out first. log( + log.0 log( log log log In equations like the following, we must obtain the value of the expression in brackets before finding its log and proceeding with the other arithmetic. 000 University of Central Lancashire Version.0 - Page

6 circle it is easy to convert from degrees to radians and vice versa. π radians degrees 0 0 degrees radians π Trigonometric functions Some areas of astronomy make use of standard trigonometric functions: sin (sine, cos (cosine and tan (tangent. All three functions are derived from the sides of a right angle triangle, which in the equations below are represented by AB, BC and AC. A BC sin A AC AB cos A AC BC tan A AB hypotenuse The angle A can be measured in either degrees or radians. Most calculators have a switch to toggle between the two units. Use a calculator to find the sine of the following angles: sin sin sin sin 0 sin Use your calculator to find the following: C B sin 0 cos 0 tan 0 sin 0 cos 0 tan Inverse trig functions Sometimes we know the value of the sine, but not the angle A. Here is an example: sin A 0. To find the angle we use the inverse sine function sin. Your calculator will have a special key to do this operation. The value A in the above example is found from: A sin 0.. You could read this to yourself as A is the angle whose sine is 0.. This might help to overcome a common misunderstanding sin 0. does not mean A sin 0. To get this expression you would write: A ( sin 0. sin 0. If we know the lengths of the appropriate sides BC and AC in the right-angled triangle we can calculate the sine of the angle A and hence the angle itself. The following assumes that the angle A is in degrees. Given info Derived info Result Side BC Side AC BC/AC Sin A A / 0..../ 0.. Now try the following examples: Side BC Side AC Sin A A. Using symbols in equations Because of the size of many numbers it is convenient to write equations using symbols. 000 University of Central Lancashire Version.0 - Page

7 Here is a simple example for the area of a circle. Area πr In this equation π ( pi is the number with approximate value. and r is the radius of a circle. If r is then: Area πr. r r.. The area may be given the symbol A so that the equation can be written: A πr Now find the area of a p coin (in units of square millimetres. Measure the diameter in millimetres using a ruler and then calculate the radius. Now use the above equation to calculate the area. Suppose you are presented with an equation of the form: L πr σ T There are two powers in this equation which could be written out in long hand: L π ( r r σ ( T T T T To calculate L (the answer all we have to do is make sure that we know the numerical value for each of the symbols to the right of the equal sign. Calculate L when T 000, r and σ.. L πr σt. ( Another straightforward example is based on the calculation of the total mass in a sphere. Let M represent the mass, r the radius of the sphere and ρ the density of material. πr M Volume density ρ The values to use are: r ρ 00 kg m π. m - Hence the mass is calculated from:. ( M kg 00 Some of the equations in the certificate courses contain expressions that are a little more complicated. The following equation describes the magnitudes of two stars: f m m.log f m, m, f and f each represents a number. The and are called subscripts. Here they are used to show which quantity refers to which star. Questions will often specify the values of all but one variable leaving you to calculate the missing value. Suppose that f 0 and f 0 so that the right hand side of the equal sign can be calculated. The variable m may also be specified. Let s suppose m. f m m. log f 0 m. log 0 m. log Note that in the third line above, the unknown m was obtained on its own on the left-hand side of the equation by adding to both sides. This follows the rules described in the next section about rearranging equations. Solve the same equation above for m for the four different sets of m, f and f given in this table. 000 University of Central Lancashire Version.0 - Page

8 m f f Answer Rearranging equations Sometimes to perform a calculation we will have to rearrange an equation to isolate the variable that we need. All equations are presented with an expression on both sides of the equal sign. To rearrange the equation we make identical operations on each expression. (Make sure you change the entire expression and not just a small part of it. For example, consider the area of a circle. Suppose that the area is known and we wish to derive the radius. This means that it would be better to write the equation with the radius alone on the left-hand side of the equal sign. The standard way of presenting the equation is as follows: A πr You are allowed to add, subtract divide or multiply the equation by anything provided that you do exactly the same to both sides. The first step is to divide both sides by pi. A πr π π A r π Now we can take the square root of both sides A r π A r π Thus the radius can be calculated from the area: r A π variable is buried deep on the right hand side of the equation. f m m. log f This will require the rearrangement of the equation, so that the unknown f appears on the left. f ( m ( m. log f ( m m f ( log. f We now need to take antilogs: ( m m. f f So we take the inverse (or turn both sides upside down which includes changing the sign of the power on the right hand side. f f f f m m. m m. Now try the following, rearranging each equation so that the variable given is alone on the left-hand side. y m x+c L π r σ T in terms of m in terms of T The same principles can be applied to other equations that we have seen. In the example above on stellar magnitude, suppose we had needed to find f. This looks difficult since the 000 University of Central Lancashire Version.0 - Page

9 000 University of Central Lancashire Version.0 - Page Answers to Problems Powers of ten 0,000 0,000, ,000, ,000 Multiplying powers of M Dividing by powers of Scientific notation Powers in general ( ( 0 ( ( ( ( ( ( ( ( ( Square roots. 0 Logarithms log log log log log

10 log 0.0 log(0.0 log(0. 0. log 000. log( log + log.0 log0.0 (alternative method log 0.0 log 0.. log.. log. log.0 Inverse logarithms log x. log x. log x. log x x, x, x,, x 0.0 Order of evaluation 0 +. log +. log log.0 + log( log +.. Trigonometric functions sin 0.0 sin 0.0 sin 0. sin cos 0.0 tan sin 0.0 cos tan.0 Inverse trig functions BC AC Sin A A Using symbols in equations Area of a p Diameter mm A πr π π. mm m b b Answer Rearranging equations y mx + c y c mx y m x L πσr So c L πr σt T / L T πσr / L πσ r / 000 University of Central Lancashire Version.0 - Page

ASTR 1030 Astronomy Lab 22 Using Your Calculator USING YOUR CALCULATOR Calculator Rule #1: Don't switch off your brain when you switch your calculator on! Your calculator is intended to eliminate tedious

### The Power of Calculator Software! Zoom Math 400 Version 1.00 Upgrade Manual for people who used Zoom Math 300

The Power of Calculator Software! Zoom Math 400 Version 1.00 Upgrade Manual for people who used Zoom Math 300 Zoom Math 400 Version 1.00 Upgrade Manual Revised 7/27/2010 2010 I.Q. Joe, LLC www.zoommath.com

### 2016 Arizona State University Page 1 of 15

NAME: MATH REFRESHER ANSWER SHEET (Note: Write all answers on this sheet and the following graph page.) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

### Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

### 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

### Remember that the information below is always provided on the formula sheet at the start of your exam paper

Maths GCSE Linear HIGHER Things to Remember Remember that the information below is always provided on the formula sheet at the start of your exam paper In addition to these formulae, you also need to learn

### Student Academic Learning Services Page 1 of 6 Trigonometry

Student Academic Learning Services Page 1 of 6 Trigonometry Purpose Trigonometry is used to understand the dimensions of triangles. Using the functions and ratios of trigonometry, the lengths and angles

### C A S I O f x T L UNIVERSITY OF SOUTHERN QUEENSLAND. The Learning Centre Learning and Teaching Support Unit

C A S I O f x - 8 2 T L UNIVERSITY OF SOUTHERN QUEENSLAND The Learning Centre Learning and Teaching Support Unit MASTERING THE CALCULATOR USING THE CASIO fx-82tl Learning and Teaching Support Unit (LTSU)

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### Examples 1, Q1 Examples 1, Q2

Chapter 1 Functions of One Variable Examples 1, Q1 Examples 1, Q2 1.1 Variables Quantities which can take a range of values (e.g. the temperature of the air, the concentration of pollutant in a river,

### 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

### AS Physics INDUCTION WORK XKCD. Student. Class 12 A / B / C / D Form

AS Physics 2014-15 INDUCTION WORK XKCD Student Class 12 A / B / C / D Form MCC 2014 1. Physical Quantities Maths and Physics have an important but overlooked distinction by students. Numbers in Physics

### CONNECT: Algebra. 3x = 20 5 REARRANGING FORMULAE

CONNECT: Algebra REARRANGING FORMULAE Before you read this resource, you need to be familiar with how to solve equations. If you are not sure of the techniques involved in that topic, please refer to CONNECT:

### C A S I O f x - 8 2 A U UNIVERSITY OF SOUTHERN QUEENSLAND. The Learning Centre Learning and Teaching Support Unit

C A S I O f x - 8 2 A U UNIVERSITY OF SOUTHERN QUEENSLAND The Learning Centre Learning and Teaching Support Unit Mastering the calculator using the Casio fx-82au Learning and Teaching Support Unit (LTSU)

### Mathematical Notation and Symbols

Mathematical Notation and Symbols 1.1 Introduction This introductory block reminds you of important notations and conventions used throughout engineering mathematics. We discuss the arithmetic of numbers,

### 11 Trigonometric Functions of Acute Angles

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

### Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS

Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS 1. Introduction and the three basic trigonometric functions (simple) Triangles are basic geometric shapes comprising three straight lines,

### C A S I O f x - 8 2 M S

C A S I O f x - 8 2 M S MASTERING THE CALCULATOR USING THE CASIO fx-82ms Learning and Teaching Support Unit (LTSU) The Learning Centre Guide book Written by Linda Galligan Published by University of Southern

### Math Analysis. A2. Explore the "menu" button on the graphing calculator in order to locate and use functions.

St. Mary's College High School Math Analysis Introducing the Graphing Calculator A. Using the Calculator for Order of Operations B. Using the Calculator for simplifying fractions, decimal notation and

### Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

### KS4 Curriculum Plan Maths FOUNDATION TIER Year 9 Autumn Term 1 Unit 1: Number

KS4 Curriculum Plan Maths FOUNDATION TIER Year 9 Autumn Term 1 Unit 1: Number 1.1 Calculations 1.2 Decimal Numbers 1.3 Place Value Use priority of operations with positive and negative numbers. Simplify

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

Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain

### Biggar High School Mathematics Department. National 4 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 4 Learning Intentions & Success Criteria: Assessing My Progress Expressions and Formulae Topic Learning Intention Success Criteria I understand this Algebra

### 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

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### S H A R P E L W H UNIVERSITY OF SOUTHERN QUEENSLAND. The Learning Centre Learning and Teaching Support Unit

S H A R P E L - 5 3 1 W H UNIVERSITY OF SOUTHERN QUEENSLAND The Learning Centre Learning and Teaching Support Unit MASTERING THE CALCULATOR USING THE SHARP EL-531WH Learning and Teaching Support (LTSU)

### Answer: = π cm. Solution:

Round #1, Problem A: (4 points/10 minutes) The perimeter of a semicircular region in centimeters is numerically equal to its area in square centimeters. What is the radius of the semicircle in centimeters?

### Linearizing Equations Handout Wilfrid Laurier University

Linearizing Equations Handout Wilfrid Laurier University c Terry Sturtevant 1 2 1 Physics Lab Supervisor 2 This document may be freely copied as long as this page is included. ii Chapter 1 Linearizing

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

GENERAL COMMENTS Grade 12 Pre-Calculus Mathematics Achievement Test (January 2015) Student Performance Observations The following observations are based on local marking results and on comments made by

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### Volume and Surface Area of a Sphere

Volume and Surface rea of a Sphere Reteaching 111 Math ourse, Lesson 111 The relationship between the volume of a cylinder, the volume of a cone, and the volume of a sphere is a special one. If the heights

### SOLVING TRIGONOMETRIC EQUATIONS

Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### 29 Wyner PreCalculus Fall 2016

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

### y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

### MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common

### Transposition of formulae

Transposition of formulae In mathematics, engineering and science, formulae are used to relate physical quantities to each other. They provide rules so that if we know the values of certain quantities,

### CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that

### T E X A S T I X I I B UNIVERSITY OF SOUTHERN QUEENSLAND. The Learning Centre Learning and Teaching Support Unit

T E X A S T I - 3 0 X I I B UNIVERSITY OF SOUTHERN QUEENSLAND The Learning Centre Learning and Teaching Support Unit TI-30XIIB (battery powered) and TI-30XIIS (solar battery powered) Learning and Teaching

### REVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit

### 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

### Trigonometry I. MathsStart. Topic 5

MathsStart (NOTE Feb 0: This is the old version of MathsStart. New books will be created during 0 and 04) Topic 5 Trigonometry I h 0 45 50 m x MATHS LEARNING CENTRE Level, Hub Central, North Terrace Campus

### SAT 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

### Time Topic What students should know Mathswatch links for revision

Time Topic What students should know Mathswatch links for revision 1.1 Pythagoras' theorem 1 Understand Pythagoras theorem. Calculate the length of the hypotenuse in a right-angled triangle. Solve problems

### Trigonometry Hard Problems

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

### PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our

### 1MA0/3H Edexcel GCSE Mathematics (Linear) 1MA0 Practice Paper 3H (Non-Calculator) Set C Higher Tier Time: 1 hour 45 minutes

1MA0/H Edexcel GCSE Mathematics (Linear) 1MA0 Practice Paper H (Non-Calculator) Set C Higher Tier Time: 1 hour 45 minutes Materials required for examination Ruler graduated in centimetres and millimetres,

### MATHEMATICS SUBJECT 4008/4028

MATHEMATICS PAPER 2 SUBJECT 4008/4028 GENERAL COMMENTS General performance of candidates was poor with the majority in the 0-10 marks group. This shows that candidates were not prepared thoroughly for

### Math Placement Test Practice Problems

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

### National 5 Mathematics Revision Notes

National 5 Mathematics Revision Notes Last updated April 015 Use this booklet to practise working independently like you will have to in the exam. Get in the habit of turning to this booklet to refresh

### 1.6 Powers of 10 and standard form Write a number in standard form. Calculate with numbers in standard form.

Unit/section title 1 Number Unit objectives (Edexcel Scheme of Work Unit 1: Powers, decimals, HCF and LCM, positive and negative, roots, rounding, reciprocals, standard form, indices and surds) 1.1 Number

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### 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

### Basic Electrical Theory

Basic Electrical Theory Mathematics Review PJM State & Member Training Dept. Objectives By the end of this presentation the Learner should be able to: Use the basics of trigonometry to calculate the different

### Of the Usage of the Sliding Rule

Of the Usage of the Sliding Rule Laurent GRÉGOIRE August 2007 ontents 1 Introduction 2 2 Mathematical principles 2 3 Multiplying 3 3.1 Principle................................. 3 3.2 and?................................

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

M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides

### Mathematics A Paper 2 (Calculator)

Write your name here Surname Other names Edexcel GCSE Centre Number Mathematics A Paper 2 (Calculator) Monday 4 March 2013 Morning Time: 1 hour 45 minutes Candidate Number Higher Tier Paper Reference 1MA0/2H

### Evaluating trigonometric functions

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

### MPE Review Section II: Trigonometry

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

### MATH LEVEL 1 ARITHMETIC (ACCUPLACER)

MATH LEVEL ARITHMETIC (ACCUPLACER) 7 Questions This test measures your ability to perform basic arithmetic operations and to solve problems that involve fundamental arithmetic concepts. There are 7 questions

### Definition III: Circular Functions

SECTION 3.3 Definition III: Circular Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Evaluate a trigonometric function using the unit circle. Find the value of a

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Chapter 6 Trigonometric Functions of Angles

6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

### 1.3 Displacement in Two Dimensions

1.3 Displacement in Two Dimensions So far, you have learned about motion in one dimension. This is adequate for learning basic principles of kinematics, but it is not enough to describe the motions of

### EQ: How can regression models be used to display and analyze the data in our everyday lives?

School of the Future Math Department Comprehensive Curriculum Plan: ALGEBRA II (Holt Algebra 2 textbook as reference guide) 2016-2017 Instructor: Diane Thole EU for the year: How do mathematical models

### Higher. Functions and Graphs. Functions and Graphs 18

hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

### 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

### how to use dual base log log slide rules

how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102

### Dyffryn School Ysgol Y Dyffryn Mathematics Faculty

Dyffryn School Ysgol Y Dyffryn Mathematics Faculty Formulae and Facts Booklet Higher Tier Number Facts Sum This means add. Difference This means take away. Product This means multiply. Share This means

### 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

### Unit 5 Trigonometry (20%)

Unit 5 (0%) Outcomes SCO: In this course students will be expected to D4 solve problems using the sine, cosine and tangent ratios B4 use the calculator correctly and efficiently C8 solve simple trigonometric

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

### (Mathematics Syllabus Form 3 Track 3 for Secondary Schools June 2013) Page 1 of 9

(Mathematics Syllabus Form 3 Track 3 for Secondary Schools June 2013) Page 1 of 9 Contents Pages Number and Applications 3-4 Algebra 5-6 Shape Space and Measurement 7-8 Data Handling 9 (Mathematics Syllabus

### Vectors and Phasors. A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles. Owen Bishop

Vectors and phasors Vectors and Phasors A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles Owen Bishop Copyrught 2007, Owen Bishop 1 page 1 Electronics Circuits

### Specimen paper MATHEMATICS HIGHER TIER. Time allowed: 2 hours. GCSE BITESIZE examinations. General Certificate of Secondary Education

GCSE BITESIZE examinations General Certificate of Secondary Education Specimen paper MATHEMATICS HIGHER TIER 2005 Paper 1 Non-calculator Time allowed: 2 hours You must not use a calculator. Answer all

### Year 1 Maths Expectations

Times Tables I can count in 2 s, 5 s and 10 s from zero. Year 1 Maths Expectations Addition I know my number facts to 20. I can add in tens and ones using a structured number line. Subtraction I know all

### 2014 Assessment Report. Mathematics and Statistics. Calculus Level 3 Statistics Level 3

National Certificate of Educational Achievement 2014 Assessment Report Mathematics and Statistics Calculus Level 3 Statistics Level 3 91577 Apply the algebra of complex numbers in solving problems 91578

### Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics.

MATH19730 Part 1 Trigonometry Section2 Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics. An angle can be measured in degrees

### Triangle Definition of sin and cos

Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ

### Precalculus, Quarter 2, Unit 2.2 Use the Unit Circle and Right Triangle Trigonometry to Find Special Angle Ratios. Overview

Use the Unit Circle and Right Triangle Trigonometry to Find Special Angle Ratios Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Use arc length of the unit circle to

### Trigonometric Identities and Conditional Equations C

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

### Paper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Thursday 5 November 2009 Morning Time: 1 hour 45 minutes

Centre No. Candidate No. Paper Reference 1 3 8 0 3 H Surname Signature Paper Reference(s) 1380/3H Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Higher Tier Thursday 5 November 2009 Morning

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

### The integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself.

Exponents An integer is multiplied by itself one or more times. The integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself. Example:

### Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015 Mathematics Without Numbers The

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

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

### 2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

### TRIGONOMETRY 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

### FX 260 Training guide. FX 260 Solar Scientific Calculator Overhead OH 260. Applicable activities

Tools Handouts FX 260 Solar Scientific Calculator Overhead OH 260 Applicable activities Key Points/ Overview Basic scientific calculator Solar powered Ability to fix decimal places Backspace key to fix

### Name Class Date 1 E. Finding the Ratio of Arc Length to Radius

Name Class Date 0- Right-Angle Trigonometry Connection: Radian Measure Essential question: What is radian, and how are radians related to degrees? CC.9 2.G.C.5 EXPLORE Finding the Ratio of Arc Length to

### Trigonometry for AC circuits

Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

### Trigonometry for AC circuits

Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

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

1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive,

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,