gc-1.ppt

AVIATION MATHEMATICS AVIATION MATHEMATICS (GC_1)(GC_1)

COURSE OBJECTIVECOURSE OBJECTIVE

Students will get an overview of aviation Students will get an overview of aviation mathematics as permathematics as per– The requirement of regulatory bodiesThe requirement of regulatory bodies– The application of mathematical concepts The application of mathematical concepts

the fieldthe field

ALLOTTED TIME AND DELIVERYALLOTTED TIME AND DELIVERY

Duration Duration – 40 hours theory40 hours theory

DeliveryDelivery– Lecture discussionLecture discussion– Class exerciseClass exercise– Reading and class exercisesReading and class exercises– Home take exams/exercisesHome take exams/exercises

COURSE CONTENTCOURSE CONTENT

ArithmeticArithmetic Basic mathematical operation Basic mathematical operation

AlgebraAlgebra Linear, simultaneous and quadratic equationLinear, simultaneous and quadratic equation

GeometryGeometry An introductory viewAn introductory view

TrigonometryTrigonometry Practical problems on charts and graphsPractical problems on charts and graphs

TEXT BOOKS AND REFERENCESTEXT BOOKS AND REFERENCES

Ac – 65 – 9A, Airframe and Powerplant Ac – 65 – 9A, Airframe and Powerplant Series, General HandbookSeries, General Handbook

Technical Mathematics with CalculusTechnical Mathematics with Calculus Shop MathematicsShop Mathematics

EVALUATIONEVALUATION

Class testsClass tests AssignmentsAssignments Final testFinal test Passing markPassing mark

– 70%70%

DISCIPLINEDISCIPLINE

PunctualityPunctuality Good appearanceGood appearance I'D. cards in proper placeI'D. cards in proper place School regulationSchool regulation

ArithmeticArithmetic

ObjectiveObjective– Addition, subtraction, multiplication and Addition, subtraction, multiplication and

division of:division of: FractionsFractions DecimalsDecimals

– Conversion of Metric System to British Conversion of Metric System to British SystemSystem

– Calculation of ratio, average and percentageCalculation of ratio, average and percentage

Basic OperationsBasic Operations

Addition +Addition + Subtraction -Subtraction - Multiplication x () Multiplication x () ** Division Division ÷,/,÷,/, Grouping signsGrouping signs

TermsTerms

NumberNumber SumSum MinuendMinuend SubtrahendSubtrahend DifferenceDifference MultiplicandMultiplicand MultiplierMultiplier ProductProduct

Terms (Contd.)Terms (Contd.)

DividendDividend DivisorDivisor QuotientQuotient RemainderRemainder DigitsDigits DenominatorDenominator NumeratorNumerator

5 – 2 =3

7 x 3 = 21

27 / 5 = 5 and 2

Product

Quotient

Difference Subtrahend

Multiplier

Minuend

Multiplicand

Divisor

Dividend

Remainder

Number SystemNumber System

Counting NumbersCounting Numbers– { 1,2,3,4,…}{ 1,2,3,4,…}

Whole NumbersWhole Numbers– { 0,1,2,3,4,…}{ 0,1,2,3,4,…}

Integers (I)Integers (I)– {…,-3,-2,-1,0,1,2,3,…}{…,-3,-2,-1,0,1,2,3,…}

Rational Numbers Rational Numbers

Fractions Fractions

a/b , a a/b , a ЄЄ I, b I, b ЄЄ I I– Proper , a<b Proper , a<b 1/21/2– Improper, a>b Improper, a>b 4/34/3

– Mixed , a c/b Mixed , a c/b 33 2/3 2/3

Decimals , 0.5, 2.33, 4.1111…Decimals , 0.5, 2.33, 4.1111… Irrational numbers , 3.030030003…, Irrational numbers , 3.030030003…, ππ Real Numbers = R U IRReal Numbers = R U IR

Significant DigitsSignificant Digits

Measured dataMeasured data Reliability of a numberReliability of a number

– Precision Precision position of last reliable digitposition of last reliable digit– Accuracy Accuracy number of significant figure number of significant figure

E.g. 56.78, 0.0034, 5.600, 3.0080, 50,000E.g. 56.78, 0.0034, 5.600, 3.0080, 50,000 Rounding off a numberRounding off a number

– Even and odd caseEven and odd case

Rules Rules

Non-zero digits are always significant. Non-zero digits are always significant. Any zeros between two significant Any zeros between two significant

digits are significant. digits are significant. A final zero or trailing zeros in the A final zero or trailing zeros in the

decimal portion decimal portion ONLYONLY are significant. are significant. Round the final result to the least Round the final result to the least

number of significant figures of any one number of significant figures of any one term. term.

Multiples and FactorsMultiples and Factors

Factors Factors 27 : 1,3,9,2727 : 1,3,9,27

MultipleMultiple 3 : 3,6,9,12,…3 : 3,6,9,12,…

Prime factorsPrime factors 36 : 2,336 : 2,3

Greatest common factor (GCF)Greatest common factor (GCF) Least common multiple (LCM)Least common multiple (LCM)

Exercise Exercise

3 + 4 – 2 x 5 + 4 3 + 4 – 2 x 5 + 4 = =

5 + 1/100 + 7/1000 = 5 + 1/100 + 7/1000 = Change 3.333 to fractional formChange 3.333 to fractional form Change 4/3 to decimal formChange 4/3 to decimal form Go to drill for significant Go to drill for significant figuresfigures

Exercises (Cont.) Exercises (Cont.)

Round off the result of the following Round off the result of the following calculations to three significant digitscalculations to three significant digits

2.4x6.5x10.372.4x6.5x10.37 21.3x0.054/(97.4x3.80)21.3x0.054/(97.4x3.80)

Find the GCF of the followingFind the GCF of the following 10,15,3010,15,30 18,30,12,4218,30,12,42

Find the LCM of the followingFind the LCM of the following 3,4,53,4,5

Measurement Systems Measurement Systems

Metric system (SI)Metric system (SI)– MeterMeter– KilogramKilogram– secondsecond

British system (BS)British system (BS)– InchInch– PoundPound– Second Second

Comparison Comparison

Ratio : by dividing one number by anotherRatio : by dividing one number by another 15 to 3 15 to 3 15:3=15/3=5 15:3=15/3=5

Proportion : equality of two ratiosProportion : equality of two ratios a/b = c/d a/b = c/d 15:3::25:515:3::25:5

Variation : the result one when the other Variation : the result one when the other changeschanges– Direct Direct – Inverse Inverse

Percentage and AveragePercentage and Average

Percentage : by the hundredPercentage : by the hundred 2 = 200%, 1.5 = 150%2 = 200%, 1.5 = 150% 50 = 25% of 40050 = 25% of 400 15% of 60 = 915% of 60 = 9

Average : Average : – Average of 3,4,5,6,7 is (3+4+5+6+7)/5 = 5Average of 3,4,5,6,7 is (3+4+5+6+7)/5 = 5

RateRate– Division by timeDivision by time

Rate

Powers and RootsPowers and Roots

Power = root Power = root exponentexponent 9 = 39 = 322

Root = Root = indexindex√ Power√ Power 3 = 3 = 33√27√27

RulesRules– aaxx a ayy = a = a x+yx+y

– aaxx/a/ayy = a = ax-yx-y

– (a(axx))yy = a = axyxy

– 1/a1/axx = a = a-x-x

– xx√a = a√a = a1/x1/x

Logarithms Logarithms

100 = 10100 = 1022

– 2 is the logarithm of 100 on the base 102 is the logarithm of 100 on the base 10

Log(ab) = loga + logbLog(ab) = loga + logb Log(a/b) = log(a) – log(b) Log(a/b) = log(a) – log(b) Log(aLog(ab)b) = b = b**log(a)log(a)

43 x 69 = x 43 x 69 = x use logarithm tables to solve use logarithm tables to solve

Algebra Algebra

Objective : Objective : – To do algebraic operationsTo do algebraic operations– To solve linear equations, simultaneous To solve linear equations, simultaneous

equations, and quadratic equationsequations, and quadratic equations

Algebraic OperationAlgebraic Operation

Algebra : Relations and properties of Algebra : Relations and properties of numbers by means of letters, signs of numbers by means of letters, signs of operations and other symbols.operations and other symbols.

3x + 4y 3x + 4y

Expression

Coefficient

Term

Laws Laws

Associative lawAssociative law 3a + (2b – 3c) = (3a +2b) – 3c 3a + (2b – 3c) = (3a +2b) – 3c (a x b) x c = a x (b x c) (a x b) x c = a x (b x c)

Commutative lawCommutative law 3a x 2b = 2b x 3a3a x 2b = 2b x 3a

Distributive law Distributive law a(b + c) = ab + aca(b + c) = ab + ac

Special ProductsSpecial Products

(a + b) (a + b) = a(a + b) (a + b) = a22 +2ab + b +2ab + b2 2

(a - b) (a - b) = a(a - b) (a - b) = a22 - 2ab + b - 2ab + b22

(a + b) (a - b) = a(a + b) (a - b) = a22 - b - b22

(a + b) (a + ) = a(a + b) (a + ) = a22 +a(b + c) + bc +a(b + c) + bc (a + b) (c + d) = ac + ad + bc + bd(a + b) (c + d) = ac + ad + bc + bd aa33 + b + b33= (a + b) (a= (a + b) (a22 - ab + b - ab + b2)2)

aa33 - b - b33= (a - b) (a= (a - b) (a22 + ab + b + ab + b2)2)

Simplification of ExpressionsSimplification of Expressions

Exercises Exercises

Equations Equations

Linear equationsLinear equations3x + 5 = 9x – 73x + 5 = 9x – 7

Word problemsWord problems Simultaneous equationsSimultaneous equations

Algebraic sentence

Quadratic EquationsQuadratic Equations

axax22 + bx + c = 0 + bx + c = 0 SolutionsSolutions

– By plotting graphsBy plotting graphs– By completing the squareBy completing the square– By quadratic formulaBy quadratic formula

a

acbbx

2

42

Geometry Geometry

Objective : Objective : – To evaluate the areas and volumes of different To evaluate the areas and volumes of different

geometric shapes.geometric shapes.– To understand the relationship of angular, linear To understand the relationship of angular, linear

and irregular geometric figures.and irregular geometric figures.

Area And VolumeArea And Volume

A = BH

A = BH

A = BH/2

A = πR2

V = πR2H

V = BHD

V = πR2H/3

V = 4πR3/3

A = 4πR2

Fundamental ConceptsFundamental Concepts

Point Point

– Designation Designation .., + , x, , + , x, ○○

Line Line – One dimensionalOne dimensional– Path traced by a pointPath traced by a point– Types Types

SegmentSegment StraightStraight curvedcurved

Plane Plane – Two dimensionalTwo dimensional– Path traced by a linePath traced by a line

Volume Volume – Three dimensionalThree dimensional– Path traced by surfacesPath traced by surfaces

Fundamental Concepts (Contd.)Fundamental Concepts (Contd.)

Angles Angles Made by two straight lines which are Made by two straight lines which are

intersectingintersecting– AcuteAcute– ObtuseObtuse– Right Right

MeasurementMeasurement– DegreeDegree– RadianRadian– GradientGradient– Revolutions Revolutions

Triangles Triangles

Right Right

Isosceles Isosceles

Equilateral Equilateral

ScaleneScalene

Polygons Polygons

SquareSquare

PentagonPentagon

HexagonHexagon

Heptagon …Heptagon …

Circles and ArcsCircles and Arcs

Area = Area = ππrr22 ,perimeter = 2 ,perimeter = 2ππr r

ArithmeticsArithmetics 3939

H

B A = 1/2 BH

AREAAREA

1. TRIANGLE


S

S

A = S2

B. RECTANGLE

B2

AREAAREA

H

B1

2. QUADRILATERALA. SQUARE

H

B

A = BH

H

B

A = BHA.TRAPEZOID

A = 1/2 ( B1 + B2) H

C. PARALLELOGRAM


AREAAREA

R

CIRCLE

A = R2 A = R2 360

SECTOR


VOLUMEVOLUME

H

W

L

S

SS

HR

CUBE RECTANGULAR BLOCK

CIRCULAR CYLINDER

V = R2HV = HLWV = S3


VOLUMEVOLUME

H

R

R

H R

CONEFRUSTUM OF A CONE SPHERE

V = 1 R2 H 3

V = 1 H(R12 +R2

2+R1R2 ) 3

V = 4 R3 3

ALGEBRAALGEBRA

EXPRESSEXPRESS ANALYZEANALYZE

EG. EG.

POWER = F ( FUEL, RPM) POWER = F ( FUEL, RPM)

ALGEBRAIC EXPRESSIONALGEBRAIC EXPRESSION

3A = 3 X A3A = 3 X A 5B + 2C = 5 X B + 2 X C5B + 2C = 5 X B + 2 X C

TERMTERM: PARTS OF EXPRESSION CONNECTED BY : PARTS OF EXPRESSION CONNECTED BY ADDITION.ADDITION.

EG. EG. 6X6X + + 5Y5Y 6X AND 5Y ARE TERMS OF THE EXPRESSION6X AND 5Y ARE TERMS OF THE EXPRESSION

COEFFICIENTCOEFFICIENT: NUMERICAL PART OF A TERM.: NUMERICAL PART OF A TERM.EG. EG. 66X + X + 55YY 6 & 5 ARE COEFFICIENTS.6 & 5 ARE COEFFICIENTS.

RULES OF ALGEBRAIC RULES OF ALGEBRAIC EXPRESSIONEXPRESSION

1. ASSOCIATIVE LAW1. ASSOCIATIVE LAW– 2C + 4D + 3F = (2C + 4D) + 3F2C + 4D + 3F = (2C + 4D) + 3F

= 2C + (4D + 3F)= 2C + (4D + 3F)

– 2C X 4D X 3F = 2C X (4D X 3F)2C X 4D X 3F = 2C X (4D X 3F) = (2C X 4D) X 3F= (2C X 4D) X 3F = 24CDF= 24CDF

2. COMMUTATIVE LAW2. COMMUTATIVE LAW– 2C + 4D = 4D + 2C2C + 4D = 4D + 2C

– 2C X 4D = 4D X 2C = 4CD2C X 4D = 4D X 2C = 4CD

RULES OF ALGEBRAIC RULES OF ALGEBRAIC EXPRESSIONEXPRESSION

3. DISTRIBUTIVE LAW3. DISTRIBUTIVE LAW– 2 (3 + 4) = 2X3 + 2X4 = 142 (3 + 4) = 2X3 + 2X4 = 14– A ( B + C ) = AB + ACA ( B + C ) = AB + AC– (A + B) / C = A / C + B / C(A + B) / C = A / C + B / C– A ( B - C ) = AB - ACA ( B - C ) = AB - AC– (A - B) / C = A / C - B / C(A - B) / C = A / C - B / C

ALGEBRAIC ADDITIONALGEBRAIC ADDITION

LIKE TERMSLIKE TERMS: TERMS THAT HAVE THE SAME : TERMS THAT HAVE THE SAME SYMBOLIC PART.SYMBOLIC PART.

TO ADD:TO ADD:– COLLECT LIKE TERMSCOLLECT LIKE TERMS– ADD COEFFICIENTADD COEFFICIENT

Eg. Eg. 3A + 5A + 9A = (3 + 5 + 9)A3A + 5A + 9A = (3 + 5 + 9)A

= = 17A17A

ALGEBRAIC MULTIPLICATIONALGEBRAIC MULTIPLICATION

FACTORS: FACTORS: PARTS OR ELEMENT PARTS OR ELEMENT SYMBOLS OPERATED BY SYMBOLS OPERATED BY MULTIPLICATION.MULTIPLICATION.

TO MULTIPLY:TO MULTIPLY:– COLLECT FACTORSCOLLECT FACTORS

EG. 2 X B X C = 2BCEG. 2 X B X C = 2BC

CONVENTIONCONVENTION

BODMASBODMAS = ( BRACKET OF DIVISION, = ( BRACKET OF DIVISION, MULTIPLICATION, ADDITION, AND MULTIPLICATION, ADDITION, AND SUBTRACTION)SUBTRACTION)

EG. EG. – A + B X C = A + BCA + B X C = A + BC– (A + B) X C = AC + BC(A + B) X C = AC + BC

SYMBOLS OF GROUPINGSYMBOLS OF GROUPING– ( ) , [ ] , { }( ) , [ ] , { }

1. a1. ann = a. a. a. … . a (to n factors of “a”) = a. a. a. … . a (to n factors of “a”) 2 a2 amm . a . ann = a = am+nm+n

(a (a mm))nn = a = amnmn

(ab)(ab)nn = a = ann.b.bnn

(a/b)(a/b)nn = a = ann/b/bnn

(1/b)(1/b)nn= 1/b= 1/bn n =b=b-n-n

aamm/a/an n = a= a(m-n)(m-n)

- - - - - {If n is even (any - - - - - {If n is even (any integer) integer) then a>o if n is odd then a>o if n is odd then athen aR.}R.}

aa00 = 1, a = 1, a00

Rules of ExponentRules of Exponent

nn aa /1

Special Products and FactorsSpecial Products and Factors

aa22 - b - b22 = (a + b) (a – b) = (a + b) (a – b) aa22+2ab+b+2ab+b22 = (a+b) (a+b) = (a+b) = (a+b) (a+b) = (a+b)22

aa22– 2ab + b– 2ab + b22 = (a-b) (a–b) = (a- b) = (a-b) (a–b) = (a- b)22

aa33 – b – b33 = (a- b) (a = (a- b) (a22 + ab + b + ab + b22)) aa33 + b + b33 = (a+b) (a = (a+b) (a22 – ab + b – ab + b22))

EquationEquation

Expression related to each other by an equality sign (=) Expression related to each other by an equality sign (=) Eg. 2x2 + 4x +3 = 7x + 5Eg. 2x2 + 4x +3 = 7x + 5 x+5y =2y+3xx+5y =2y+3x

Solving Linear EquationSolving Linear Equation Add or subtract the same number on both sides to Add or subtract the same number on both sides to

collect the same terms to one side.collect the same terms to one side. Multiply or divide both sides by the same number to Multiply or divide both sides by the same number to

solve for the variable.solve for the variable.Eg. 3x + 5 = 2x + 7Eg. 3x + 5 = 2x + 7Step 3x + 5 – 5 = 2x + 7 – 5Step 3x + 5 – 5 = 2x + 7 – 5 3x = 2x + 23x = 2x + 2 3x – 2x = 2x – 2x + 23x – 2x = 2x – 2x + 2 x = 2x = 2

Quadratic EquationQuadratic Equation

These are equation of second order.These are equation of second order.

Quadratic equation in one variable.Quadratic equation in one variable.3x3x22 + 5x + 2 = 0 + 5x + 2 = 0 axax22 + bx + c = 0 , + bx + c = 0 , aa00

Quadratic equation in two variable Quadratic equation in two variable axax22 + bx +c + dy + bx +c + dy22 + ey + f = 0 + ey + f = 0

Where a, b, c, d, e, and f are constants. a and d are Where a, b, c, d, e, and f are constants. a and d are different from zero.different from zero.

Consider:Consider:axax22 + bx + c = 0 , + bx + c = 0 , aa00

Case 1: When b = 0Case 1: When b = 0 axax22 + C = 0 + C = 0Solving for xSolving for x xx22 = = -c -c aa

Solving Quadratic EquationSolving Quadratic Equation

)(a

cx

,

0a

c

Case 2: When c = 0Case 2: When c = 0

axax22 + bx = 0 + bx = 0

Solving for x:Solving for x:

axax22 + bx = 0 + bx = 0

x (ax + b ) = 0x (ax + b ) = 0

x = 0x = 0 or ax + b = 0 or ax + b = 0

ax = -bax = -b

x = -b/ax = -b/a

Case 3:Case 3: a, b ,and c a, b ,and c 0 0

Eqn. axEqn. ax22 + bx + c = 0 + bx + c = 0

This can be solved by one of the following.This can be solved by one of the following. Plotting ( inaccurate) **Plotting ( inaccurate) ** FactorizationFactorization Completing the squareCompleting the square Quadratic formulaQuadratic formula

Quadratic Formula:

. . . .. . . . . . 2

42

a

acbbX

,

042 acb

Simultaneous EquationsSimultaneous Equations

Linear simultaneous equation can be solved byLinear simultaneous equation can be solved by Graphical method (approximate)Graphical method (approximate) Algebraic methodAlgebraic method

– Elimination Elimination – Substitution (*)Substitution (*)

Eg. Eg. 2x + 4y = 5 (1)2x + 4y = 5 (1)

x + y = 3 (2)x + y = 3 (2)

GeometryGeometry 5959

GEOMETRYGEOMETRYFundamentals of geometry1. Point .

2. Line

3. Straight line

4. Line Segment

5. Half Straight line

6. Parallel line


Complementary Angles

Suplementary angles

Adjacent Angles

Equilateral Triangle

Isosceles Triangle

Right Angle Triangle

• Oblique Angle Triangle


7. Angle

a) Straight angle

b) Obtuse

c) Right angle

d) Acute


Square

Rectangle

Parallelogram

Rhombus

Chord

Sector

Segment

Tangent to Circle

Quadrilateral Circle


RulesRules

1. Opposite or vertical angles are equal.

2. Alternate interior angles are equal.


RulesRules

3. Corresponding angles are equal.

4. The sum of the interior angles of a triangle is always 180.


RulesRules

A

D

BCF E

D

F E

A

BC

5. Two triangles are similar when their corresponding angles are equal.Corresponding sides of similar triangles are proportional

6. Two triangles are congruentI. SASII. ASAIII. SSS


RulesRules

7. Pythagoras Theorem

8. The sum of the interior angles of a convex poly gon with n-sides is = (2n – 4)rts. = 180(n - 2)

a c

b

a2 + b2 = c2


RulesRules

A B

CD

k

9. The sum of the exterior angles of a convex polygon with n sides is = 4 rts (360o).

10. If ABCD is a parallelogramI. AB = CD and AD =BC;II. A = C and B = D;III. BD bisect area ABCD.IV. AK = KC and BK = KD


11. Condition for a quadrilateral to be a parallelogramI. AB is equal and parallel to DC,II. A=C and B = DIII. If AB = DC and AD =BCIV. If AK = KC and BK=KD

12. TriangleI.11. If in ABC, AC > AB then B > C.II.If in ABC, B > C then AC > AB .III.If ABC is any triangle, AB + AC > BC

A

B C

A B

CD

k


RulesRules

B C

H K

A

N PA B

13. If CN is the perpendicular from C to a straight line AB and NP then CN < CP

14. If H, K are the mid-points of AB, AC respectively, thenHK is Parallel to BCHK = BC / 2


RulesRules

-The altitudes of a triangle are concurrent.The point is orthocentre or the triangle.

A

C

B

D

Pwere

Q

R

S

E T

15. If two transversals ABCDE, PQRST are cut by the parallel lines BQ, CR, DS, ET, and if BC=CD=DE then QR = RS = ST.

16. The medians AD, BE, CF of ABC concur at a point G, such thatDG = 1/3DAEG = 1/3 EBFG = 1/3FG

A

B

C

D

E

F

G


RulesRules

17. The perpendicular bisectors of the three sides of a triangle are concurrent. The point at which they concur is the circum-center of the triangle

18.The altitudes of a triangle are concurrent

The point is called ortho-center of the triangle


RulesRules

19.The internal bisectors of the three angles of a triangle are concurrent.

The point at which they concur is called the in-center of the triangle


RulesRules

CircleIf M is the mid-point of a chord AB of a circle, center o, then < OMA = 1rt <

If the chords AB and CD of a circle are equal, they are equidistant from the centre.

If the chords AB and CD of a circle equidistance from the centre, then AB = CD

D

C

A

Bx

OM


RulesRules

There is one circle, and only one circle that pass through There is one circle, and only one circle that pass through three given points A, B, C not in the same straight line.three given points A, B, C not in the same straight line.

The perpendicular bisectors of AB , BC, and CA meet at The perpendicular bisectors of AB , BC, and CA meet at the centre 0 of the circle.the centre 0 of the circle.

The angle which an arc of a circle subtends at the center The angle which an arc of a circle subtends at the center is double that which subtends at any point on the is double that which subtends at any point on the remaining part of the circumference remaining part of the circumference

<AOB = 2 x <ACB<AOB = 2 x <ACB

x

A

B

C

O

The end.The end.

Trigonometry Trigonometry

Sine Sine α = A/C = A/C cosine cosine α = B/C = B/C Tangent Tangent α = B/A = B/A

C

A

B

α

Charts & GraphsCharts & Graphs

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