Mathematical Concept and Techiques_final

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COURSE OUTLINE:

I. ARITHMETIC

• Basic Operation Involving Whole Numbers• Basic Operation Involving Fractions

• Basic Operation Involving Decimals

• Ratio, Proportion and Percent

• Powers and Roots

II. SYSTEMS OF MEASUREMENT

• Metric System

• Unit Conversion and Conversion Factors

III. GEOMETRICAL APPLICATION

• Points, Lines and Planes

• Angles and Triangles

• Polygons and Circles



IV. BASIC MENSURATION

• Areas and Volumes

V. LINES AND THE CARTESIAN PLANE

VI. BASIC TRIGONOMETRICAL FUNCTION



Operations with Whole Numbers

Addition represents the idea of finding a total count, or

summing up, of values. Since we use only ten digits in our

system (remember base 10), it is often necessary to use

place value to “carry” digits.

Properties of Addition





EXERCISES:

I. Perform the following additions and subtractions.

1. 2,456 + 8,946 2. 534 – 276

II. Perform the following multiplications.

1. 1,859 • 68 2. 2,695 • 465

III. Perform the following divisions.

1. 14,846 ÷ 124 2. 33,429 ÷ 132





TYPES OF FRACTIONS

1. Proper Fractions – are fractions in which the numerator is

smaller than or less than ( < ) the denominator.

Example:

2

3

,4

5

,3

8

,7

12

2. Improper Fractions – are fractions in which the

numerator is larger than or greater than ( > ) the

denominator. ( The value of an improper fraction is

always greater than one.)



Example:

5

3

,7

6

,5

4

,12

7

3. Mixed Fractions

- A simplified improper fraction.- Numbers used to express amounts that are

greater than a whole and have a fractional part.

- A combination of whole number and a proper

fraction.Example:

7

1= ;

1

61

3

54



Simplification of Fractions:

Rule: To reduce fraction to its simplest form, divide both the

numerator and denominator by a common number. If both

numerator and denominator are no longer divisible by a

common number except one ( 1 ), then it is in simplest form.

Example:

6

15=

2

5

( divide both numerator and

denominator by 3 )

Least Common Denominator:

1. In addition and subtraction of fractions, the numerator are

added or subtracted directly for common denominator.



Example:

1

3+

1

3+

2

3= =

1 + 1 + 2

3

4

3or

1

31

1

5 –

2

5+

4

5=

1 – 2 + 4

5=

3

5

Note: The numerator of fractions having the same

denominator can be added or subtracted directly.



2. In addition or subtraction of fractions whose denominatorare not the same, we find the least common denominator.

LCD – the lowest number that is exactly divisible by all the

denominators of the fractions to be added or subtracted.

To find the least common denominator ( LCD ), factor each

denominator of the given fractions and multiply all factorsto the highest exponent as it appear in any of the

denominators.



Example:

1

5+ 1

10+ 1

25

Denominators: 5 = 5 x 1

10 = 5 x 2 x 1

25 = ( 5 )2 x 1

The factors that are present ( not necessarily common ) are 5,

2 & 1

The highest exponent of 5 is square ( 2 )

The highest exponent of 2 is one ( 1 )

The highest exponent of 1 is one ( 1 )



Therefore the LCD is ( 52 x 2 x 1 ) = 50

1

5+

1

10+

1

25

=10 + 5 + 2

50

17

50

Example:

1

2+ 1

3 – 1

9



2 = 2 * 1

3 = 3 * 1

9 = 32 * 1

LCD = 2 * 32 * 1 = 18

9 + 6 – 2

18=

13

18

Denominators:



Operations of Fractions:

1. Addition of Fractions:

To add fractions of the same denominator, add the

numerator and copy the common denominator andreduce answer to its simplest form.

Example:

1

3+ 2

3= 3

3=a. 1

b.1

4+

1

4=

2

4=

1

2



To add fractions of different denominators, find the least

common denominator and add. Simplify answer to its

simplest form.

Example:

34

+a. 78

= =6 + 78

or138

58

1

2. Subtraction of Fractions:

To subtract fractions having the same denominator. Subtract

the numerators and copy the common denominator and

reduce answer to simplest form.



7

5 – a.

2

5= =

7 – 2

5

5

5

Example:

1=

3

4 – b.

1

4=

3 – 1

4=

2

4=

1

2

To subtract fractions having different denominators, find the

least denominator (LCD) and subtract the numerators, and

reduce final answer to simplest form.

Example:3

5

– a.1

4

= =12 – 5

20

7

20





4. Division of Fractions:

To reduce fraction by another fraction, take the reciprocal

of the divisor and then proceed to the multiplications

process. Simplify final fraction to simplest form.

Example:1

2 – a.

1

4=

1

2*

4

1= or

4

22

3

4 – b. 2 =

3

4*

1

2=

3

8



ROUNDING OFF NUMBERS

1. Round to the nearest Tenths

How do round to the nearest tenths? Follow the steps outlined below.

1. First find the tenths digit. This the first digit after the decimal place.

2.241 (Ex.1) | 4.567 (Ex.2)The tenths digit is shown using the red arrow.

2. Now, find the next digit to the right.

2.241 (Ex.1) | 4.567 (Ex.2)

The next digit is shown using the green arrow.

3. We round up or down depending on the value of the next digit.

(a) If the next digit is less than 5, round down. To round down,

remove all the digits beginning at the next digit.

In Ex 1., the number to the nearest tenths is 2.2.

BASIC OPERATIONS INVOLVING DECIMALS




(b) If the next digit is greater than or equal to 5, round up. Toround up, remove all digits beginning at the next digit, and add

1 to the tenths digit. In Ex 2., the number 4.567 rounded to the

nearest tenths is 4.6.

Here are two more examples.

Example 1. Round each of the two numbers to the nearest tenths.

(a) 235.233

Solution: First find the tenths digit, 235.233. Now find the next digit to

the right , 235.233, the next digit is a 3, which is less than 5, so we

round down. The answer

(b) 45.581

Solution: First find the tenths digit, 45.581. Now find the next digit to

the right, 45.581, the next digit is an 8, which is greater than 5, so we

round up. The answer is

235.2

45.6






(b) If the next digit is greater than or equal to 5, round up. To round up,remove all digits beginning at the next digit, and add 1 to the

hundredths digit. In Ex 2, the number 4.567 rounded to the nearest

hundredths is 4.56.

Example 2. Round each of the two numbers to the nearest hundredths.

(a) 17.2363

Solution: First find the hundredths digit, 17.2363. Now find the next digit to

the right, 17.2363, the next digit is a 6, which is greater than 5, so we

rounded up. Remove all digits beginning with the next digit and increase

the hundredths digit by one. The answer

(b) 54.581

Solution: Find the hundredths digit, 54.581. Now find the next digit to the

right, 54.581, a 1, which is less than 5, so we round down.

The answer

17.24

54.58




3. Round to nearest Whole Number

How do you round to the nearest whole number? This is also called rounding to

the nearest units. Follow the steps outlined below.

1. First find the units digit, the first digit before the decimal place.

2.241 (Ex.1) | 4.567 (Ex.2)The units digit is shown using the red arrow.

2. Now, find the next digit to the right .

2.241 (Ex.1) | 4.567 (Ex.2)

The next digit is shown using green arrow.



remove all digits beginning at the next digit. In Ex 1, the number 2.241

rounded to the nearest units (or whole number) is 2.

(b) If the next digit is greater than or equal to 5, round up.




To round up, remove all digits beginning at the next digit, and add 1 tothe units digit. In Ex 2, the number 4.567 rounded to the nearest units ( whole

number ) is 5.

Example 3. Round each of the two numbers to the nearest whole number ( units ).

(a) 17.2363

Solution: First find the units digit, 17.2363. Now find the next digit to the right,

17.2363, the next digit is a 2, which is less than 5, so we round down. The

answer .

(b) 54.581

Solution: First find the units digit, 54.581. Now find the next digit to the right,

54.581, the next digit is 5, which is greater than 5, so we round up. The

answer .

17

55




4. Round to nearest Tens

How do you round to the nearest tens? Follow the steps outlined below.

1. First find the tens digit, the second digit before the decimal place.

62.64 (Ex 1) | 27.56 (Ex 2)

The tens digit is shown using the red arrow.


62.64 (Ex 1) | 27.56 (Ex 2)



(a) If the next digit is less than 5, round down. To round down, remove

all digits beginning at the next digit. In Ex 1, the number 62.24

rounded to the nearest tens place is 60.




(b) If the next digit is greater than or equal to 5, round up. To round up,remove all digits beginning at the next digit, and add 1 to the tens digit. In Ex

2, the number 27.65 rounded to the nearest tens place is 30.

Example 4. Round each of the two numbers to the nearest tens.

(a) 17.2363

Solution: First find the tens digit, 17.2363. Now find the next digit to the right,

17.2363, the next digit is a 7, which is greater than 5,so we round up. The

answer is

(b) 54.581

Solution: First find the tens digit, 54.581. Now find the next digit to the right,

54.581, the next digit is a 4, which is less than 5, so we round down. The

answer is

20

50




5. Round to nearest Hundreds

How do you round to the nearest hundreds? Follow the steps outlined below.

1. First find the number digit, the third digit before the decimal place.

642.24 ( Ex 1 ) | 272.56 ( Ex 2 )

The hundreds digit is shown using the red arrow.


642.24 ( Ex 1 ) | 272.56 ( Ex 2 )



(a) If the next digit is less than 5, round down. To round down, remove

all the digits beginning at the next digit. In Ex 1, the number 642.24

rounded to the nearest hundreds place is 600.




(b) If the next digit is greater than or equal to 5, round up. To round up,remove all the digits beginning at the next digit, and add 1 to the hundreds

digit. In Ex 2, the number 272.56 rounded to the nearest hundreds place is

300.

Example 5. Round each of the two numbers to the nearest hundreds.

(a) 717.23

Solution: First find the hundreds digit, 717.23. Now find the next digit to the

right, 717.23, the next digit is a 1, which is less than 5, so we round down.

The answer is

(b) 254.58

Solution: First find the hundreds digit, 254.58. Now find the next digit to the

right, 254.58, the next digit is a 5, which is greater than or equal to 5, so we

round up. The answer is

700

300




6. Round to nearest Thousands

How do you round to the nearest thousands? Follow the steps outlined below.

1. First find the thousands place, the fourth digit before the decimal place.

6242.24 ( Ex 1 ) | 2721.56 ( Ex 2 )

The thousands digit is shown using the red arrow.


6242.24 ( Ex 1 ) | 2721.56 ( Ex 2 )




remove all digits beginning at the next digit. In Ex 1, the number

6242.24 rounded to the nearest thousands place is 6000.




(b) If the next digit is greater than or equal to 5, round up. Toround up, remove all digits beginning at the next digit, and add

1 to the thousands digit. In Ex 2, the number 2721.56 rounded

to the nearest thousands place is 3000.

Example 6. Round each of the two numbers to the nearest thousands.

(a) 1717.23

Solution: First find the thousands digit, 1717.23. Now find the next digit to the

right, 1717.23, the next digit is a 7, which is greater than 5, so we round up.

The answer is

(b) 8254.58

Solution: First find the thousands digit, 8254.58. Now find the next digit to the

right, 8254.58, the next digit is a 2, which is less than 5, so we round down.

The answer is

2000

8000






(b) If the next digit is greater than or equal to 5, round up. To round up,remove all digits beginning at the next digit, and add 1 to the ten thousands

digit. In Ex 2, the number 27,213, rounded to the nearest thousands place is

30,000.

Example 7. Round each of the two numbers to the nearest ten thousands.

(a) 51,717.2

Solution: First find the ten thousands digit, 51,717.2. Now find the next digit to

the right, 51,717.2, the next digit is a 1, which is less than 5, so we round

down. The answer is

(b) 38,254

Solution: First find the ten thousands digit, 38,254. Now find the next digit to

the right, 38,254, the next digit is a 8, which is greater than 5, so we round

up. The answer is

50,000

40,000



EXERCISES: ROUNDING OFF NUMBERS

Exercises 1: Round each to the nearest tenths.

1. 2.67 3. 9.457 5. 17.232

2. 5.71 4. 12.59

Exercises 2: Round each to the nearest hundredths

1. 5.347 3. 0.136 5. 1.345

2. 12.423 4. 39.126

Exercises 3: Round each to the nearest whole numbers

1. 2.67 3. 9.457 5. 7.23

2. 13.59 4. 57.23









Ratio

A ratio is a pair of numbers that compares two quantities

or describes a rate. A ratio can be written in three ways:

Using the word to 3 to 4

Using a colon 3:4Writing a fraction 3/4

When a ratio is used to compare two different kinds of

quantities, such as miles to gallons, it is called a rate.Equal ratios make the same comparison. To find equal

ratios, multiply or divide each term of the given ratio by

the same number.

These are all read 3 to 4.



EXAMPLE:

In the shop, there are 7 MS plates and 12 Aluminum

plates. In three different ways, write the ratio of:

a. MS plates to Aluminum plates

b. Aluminum plates to MS plates

c. MS plates to pieces of plates



Proportion

A proportion is a statement that two ratios are equal. A

proportion shows that the numbers in two different ratios

compare to each other in the same way.

In a proportion, the cross products are equal. 3 and

10 are the means, or the terms in the middle of the

proportion. 2 and 15 are the extremes, or the terms atthe beginning and end of the proportion.



EXAMPLE:



Use the figure below. The small gear makes 8 turns for

every 5 turns of the large gear.

EXERCISE:

How many turns will the large gear

make if the small gear makes 80 turns?

How many turns will the small gear make

if the large gear makes 62.5 turns?



Percentage:

In machine shop and in daily life, we often need to work with

percentage. Like decimals, percents are closely related to

fractions. Decimals are a simple way of writing fractions of

tenths, hundredths, thousandths, and ten thousandths.

Similarly, percents are a simple means of writing fractions of

hundredths, Percent are also a means of expressing ratios.

In working with percents as ratio, we make a comparison

against a base of 100.



Interpretation of Percents:

Percent means “ per one hundred “. Percent written in

fractional form have 100 as the base of comparison

(denominator).

Example: 3

100

, 25

100

, 63

100

, 145

100

Fraction of hundredths can be written in decimal form:

0.03 0.25 0.63 1.45



We also can write fractions of hundredths as percents by

using symbol ( % ). Thus, in percent notation, these fractions

are written:

3% 25% 63% 145%

As ratio expressions, percents compare an amount to a base

of 100. For example 3% ( 3 per 100 ) is the same as the ratio

3 to 100.

3% =3

100= 0.03 = 3 : 100



Since percents are ratio expressions, they can be used in

stating proportions. In a proportion expression, the percentratio (amount per 100) is equated with another ratio. The two

value that form the other ratio are called the percentage and

the base.

Percentage is the number expressed as a percent of another

number. The base is the value to which the percentage is

being compared. For example, if we say that 20 is 50% of

40, 20 is the percentage, 40 is the base to which it is being

compared, and 50 is the percent.

Thus, in percent proportion, the ratio of the percent to 100 is

equal to the ratio of percentage to the base.



Percent

100

=

percentage

base

Example: What percent of 75 is 15

x

100=

15

75

x =100 ( 15 )

75= 20 %



RULE:

To find the percent, multiply the percentage by 100 and

divide by the base.

Percent =100 x percentage

Base

To find the percentage, multiply the percent by the baseand divide by 100.

Percentage =percent x base

100





Operations with powers

Multiplication and division of powers.

Power of product of some factors.

Power of a quotient (fraction).

Raising of power to a power.

Negative, zero and fractional exponents of a power.

Operations with roots.

Arithmetical root.

Root of product of some factors.

Root of quotient (fraction).Raising of root to a power.

Proportional change of degrees of a root and its radicand.

About meaningless expressions.



34

The base

figure

The figure „4‟ is

the power or

„replicator‟

A power describes or states by how many times

the base figure should be multiplied by itself.



34

= 3x3x3x3 Typical economics and business examples where

powers can be used include:

- calculating compounded figures such asinterest payments on a mortgage;

- working out areas or volumes;

- estimating the expected return from an

investment project (net present values);and

- calculating the value of an individual‟s

savings or pension in times of high and

sustained inflation.



1. At multiplying of powers with the same base their exponentsare added:

a m · a n = a m + n .

2. At dividing

of powers with the same base their exponents are

subtracted:

3. A power of product of two or some factors is equal to a

product of powers of these factors:

( abc… ) n = a n · b n · c n …



4. A power of a quotient (fraction) is equal to a quotient of

powers of a dividend (numerator) and a divisor (denominator):

( a / b ) n = a n / b n .

5. At raising of a power to a power their exponents are

multiplied:

( a

m

)

n

= a

m

n

.





Work out the following expressions using simple powers:

33 =

53 =

66 =

113 =

4.53 =

N ti t f A f b



Similarly, powers can be used in a „symmetrical way‟. A

negative power can be expressed as a positive power when

it is the reciprocal.

Example:

3-2 = 1/32 = 1/9

4-3 = 1/43 = 1/(4x4x4) = 1/64

5

-5

= 1/5

5

= 1/(5x5x5x5x5) = 1/3125

Negative exponent of a power. A power of some number

with a negative (integer) exponent is defined as unit divided

by the power of the same number with the exponent equal

to an absolute value of the negative exponent:



In all below mentioned formulas a symbolmeans an arithmetical root

( all radicands are considered here only positive ).

1. A root of product of some factors is equal to a product of roots of these factors:

2. A root of a quotient is equal to a quotient of roots of a dividend and a divisor:



3. At raising a root to a power it is sufficient to raise a radicand to this power:

4. If to increase a degree of a root by n times and to raise simultaneously its radicand to the n-

th power, the root value doesn’t

change:

5. If to decrease a degree of a root by n times and to extract simultaneously the n-th degree root

of the radicand, the root valuedoesn’t change:



Zero exponent o f a power . A power of any non-zero number

with zero exponent is equal to 1.

E x a m p l e s: 2 0 = 1, ( – 5 ) 0 = 1, ( – 3 / 5 ) 0 = 1.

Fract ional exponent of a power .

To raise a real number a to a power with an exponentm / n it is necessary to extract the n-th degree root from

the m-th power of this number a:



ENGLISH AND METRIC SYSTEMOF MEASUREMENT

Wh M t i ?



Why Metric?

A. The metric system is much easier. All metric units are

related by factors of 10.B. Nearly the entire world (95%), except the United States,

now uses the metric system. U.S. economic competitiveness

would be strengthened by converting to the metric system.

C. Metric is used exclusively in science -- therefore,understanding of scientific and technical issues by non-

scientists will be enhanced if the metric system is universally

adopted.

D. Because the metric system uses units related by factorsof ten and the types of units (distance, area, volume, mass)

are simply-related, performing calculations with the metric

system is much easier ¾ thus facilitating quantitative

analysis and understanding in science.



Comparison of simple conversion operations in the English

(customary) and Metric systems

Notice the unusual numbers relating the various units in the English system and the

simplicity of the powers of ten in Metric.











UNIT CONVERSION AND CONVERSION FACTORS

A unit conversion expresses the same property as a differentunit of measurement.

A conversion factor is a number used to change one set of

units to another, by multiplying or dividing.

When a conversion is necessary, the appropriate conversion

factor to an equal value must be used.

For example, to convert inches to feet, the appropriate

conversion value is 12 inches equal 1 foot.



A unit cancellation table is developed by using known units,

conversion factors, and the fact that a unit of measure ÷ the

same unit of measure cancels out that unit. The table is set up

so all the units cancel except for the unit desired. To cancel a

unit, the same unit must be in the numerator and in the

denominator. When you multiply across the table, the top

number will be divided by the bottom number, and the result

will be the answer in the desired units.





GEOMETRICAL APPLICATION



POINTS, LINES, AND PLANES

A Point has no dimension. It is usually represented by a

small dot.

A Line extends in one dimension. It is usually representedby a straight line with two arrowheads to indicate that the

line extends without end in two directions.

A Plane extends in two dimensions. It is usually representedby a shape that looks like a tabletop or wall.



A few basic concepts in geometry must also be commonly

understood without being defined. One such concept is the idea

that a point lies on a line or a plane.

are points that lie on the same line.

are points that lie on the same plane.

Collinear points

Coplanar points

Examp le: Nam ing Co l l inear and Coplanar Points

a. Name three points that are collinear.

b. Name four points that are coplanar.

c. Name three points that are not collinear.



Another undefined concept in geometry is the idea that a point

on a line is between two other points on the line.

Consider the line AB (symbolized by AB ).

The line segment or segment AB (symbolized by AB )

consists of the endpoints A and B, and all points on AB

that are between A and B.

A B

Line

A B

Segment



The ray AB (symbolized by AB ) consists of the initial point A

and all points on AB that lie on the same side of A as point

B.

Note that AB is the same as BA , and AB is the same as BA

. However, AB and BA are not the same. They have different

initial points and extend in different directions.

A B

Ray

A B

Ray



If C is between A and B, then CA and CB are opposite

rays

A B

Ray

C

Like points, segments and rays are collinear if they lie on

the same line. So, any two opposite rays are collinear.

Segments, rays, and lines are coplanar if they lie on the

same plane.



INTERSECTIONS OF LINES AND PLANES

Two or more geometric figures intersect if they have one ormore points in common. The intersection of the figures is the

set of points the figures have

in common.



ANGLES AND TRIANGLES



DIFFERENT TYPES OF ANGLES

ACUTE ANGLE OBTUSE ANGLE

RIGHT ANGLE



REFLEX ANGLE

STRAIGHT ANGLE



DIFFERENT TYPES OF TRIANGLES

ACUTE TRIANGLE

EQUILATERAL TRIANGLE



ISOSCELES TRIANGLE

OBTUSE TRIANGLE



RIGHT TRIANGLE

SCALENE TRIANGLE







POLYGONS AND CIRCLES



POLYGONS

A regular polygon is a polygon whose sides are all the

same length and whose interior angles all have the same

measure.

An equilateral triangle is a 3-sided regular polygon;

a square is a 4-sided regular polygon.

CIRCLE

A circle is the locus of all points equidistant from a

central point.

I i ti d Ci i ti



Inscription and Circumscription

Certain geometric figures are created by combining circles

with other geometric figures, such as polygons. There are

two simple ways to unite a circle with a polygon. One is

inscription, and the other is circumscription.

When a polygon is inscribed in a circle, it means that

each of the vertices of that polygon intersects the circle.

When a polygon is circumscribed about a circle, it

means that each of the sides of the polygon is tangentto the circle. Below these situations are pictured.



INSCRIBED CIRCUMSCRIBED

Above on the left, the hexagon ABCDEF is inscribed in

the circle G. On the right, the quadrilateral ABCD is

circumscribed about the circle E.



Concentric circles are circles that have the same center.

Just because a circle is inside another circle doesn'tmean they are concentric; they must have the same

point as their center. Any number of circles can be

concentric to one another, provided that they all share a

center.





BASIC MENSURATION



Circumference:

The Circle:

The circle has been called the perfect geometric form. It is

perhaps also the most important of all geometric forms.

Definition:

A circle is a plane curve consisting of all the points that are

the same distance from a fixed point called the center, thecommon distance of the points on the curve from the center

is called the radius. The region bounded by (enclosed

within) the circle is also often referred to as a circle.



The curve, (that is, the circle as defined)) is referred to as the

perimeter or circumference of the circle. These terms are

used both to denote the curve itself and its length.

Figure 1: Circle

A diameter of a circle is any line drawn from one side of the



y

circumference to the other passing through the center. A

diameter divides both the area and the circumference into two

equal (congruent) parts called semicircles. Any line drawn fromthe center of the circumference is called radius. (Note: the

plural form of “ radius “ is “ radii “).

Any portion of the circumference is called an arc.

A chord is a line segment connecting any two points in the

circle. In the figure as shown below, DE is the chord. The

chord is said to subtend its arc. The chord DE subtends the arc

DME. The area bounded by an arc and a chord is called a

segment; as in the figure, the area DME is a segment. The

area bounded by two radii and an arc is called a sector. A

sector is an area shaped like a wedge of pie, as in the area

BOC in figure 1.



Relationship between Diameter, Radius, and Circumference:

Rules:

The radius equals one-half the diameter, or, the diameter

equals twice the radius.

r =d

2

or d = 2r

The circumference equals the diameters times ii.

C = d ii





This may also be expressed in either of the following ways,

which are usually more convenient to use:

Rule: The area of a circle equals = P1 times the square

of the radius; or, the area of a circle equals one-fourth

times the square of the diameter.

If A = area, C = circumference, d = diameter, and

r = radius, these rules are expressed in the following

formulas:

π

π

C r

2 A =



d2

4 A =

A = r 2π or r = A

π

π

or A = 0.7854 d2 or A

0.7854d =

Example:

Find the radius of a circle whose area is 320 cm2 .

Using the formula r = A / and substituting the

given numbers,

Radius = 10.09 cm

π )

r =320

3.1416



Pythagorean Theorem:

The Pythagorean theorem is one of the most important

mathematical principles. Pythagorean theorem states that in

a right triangle, the square of the hypotenuse, which is the

longest side of a right triangle, is equal to the sum of the

squares of the other two sides of the triangle. This theorem is

only applicable to right triangles.

A right triangle is a triangle with one of its angles measures

90o .

a

b

c



The triangle in the figure is a right triangle. Side c is the

hypotenuse. Sides a and b are called legs.

Using the above figure, Pythagorean theorem can be

represented by the equation

c2 = a2 + b2

NOTE: We always need to know the measurements of two

sides of a triangle in order to use the basic formula.

RULE: To find the length of the hypotenuse when the length of

the altitude and base are known, use the formula;

c

2

= a

2

+ b

2



RULE: To find the length of one of the legs (altitude or base)

of a right triangle when the length of the hypotenuse and theother length are known, use one of the formulas;

b2 = c2 – a2 or a2 = c2 – b2

Example:

Given a right as shown below, calculate the missing

side.

a = 76 mm c = ?

b = 102 mm



Required: c = ?

Solution:

Using c2 = a2 + b2

= ( 76 )2 + ( 102 )2

= 5776 + 10.404

c = 127.2 mm

)



Shape & Perimeter:

1. Square – a kind of polygon having 4 equal sides

S

Area = s2

Perimeter = 4s



2. Hexagon – a polygon having 6 sides

da

s3

2 A = a2

d = 2 a = 1.155s

s = 0.866d

3. Triangle – a kind of polygon having three sides

A = ½ bh

Where: b = base

h = heightb

h



Units of time and angle

h = hoursmin = minutes

s = seconds

o = degrees

„ = minutes

“ = seconds

1. Time units

The unit of time is the second. A second is the 24 x 60

x 60th part of a mean solar day.

60 min

1 min = 60 s1 h =

Conclusion: The conversion factor from unit to unit is 60.



2. Angular units

The derived unit of angle is the degree. A degree isthe 360th part of the circumference of a circle.

60 „

1 „

1o =

= 60 “

Conclusion:

The conversion factor from unit to unit is 60.

Result

The difference between units of time and angle consists

only in the way they are named.



3. Summary

The conversion factor for units of time and angle is 60.

Conversion

Into the next smallest unit: x 60

Into the next largest unit: : 60

4. Example

A time of 1.48 hours was needed to grind the

clearance angle of a face-cutting miller. Calculate the

exact time ( t ) in h, min, s.



find h, min, s

given that t = 1.48 h

solution

1.48 h =

0.48 h = 0.48 * 60 = 28.8 min0.8 min = 0.8 * 60 =

preliminary consideration

next smallest unit by 60

1 h +

28 min +

48 s

1 h 28 min 48 s



Areas of regular quadrilaterals:

A = area

l = length of side

h = height of area

Note:

In the case of rectangle, the opposite sides are parallel.

1. Square

l

l

area = length * height

A = l * l

= l

2



2. Rhombus

l

l

harea = length * height

A = l * h

3. Rectangle

h

l


A = l * h



4. Parallelogram

h

l


A = l * h

5. Summary

For calculating the areas of rectangles, the formula is:

area = length x height

Note

The height is taken as being the perpendicular to the

line.



C l l ti f II



Calculation of areas II

A = area

l = length of side

h = height of area

1. Triangle

hh

l l

If we add a supplementary

area to triangle to form a

familiar rectangle, we have:length * width = 2 * A

Conclusion: l * h

2

A =

N t



a

a

h

Note

In the case of equilateral triangles,

Pythagoras‟ Theorem gives us a

height of:

a2 = h2 +a

2( )

2

a

2( )

2

h2 = a2 – =3

4

a2

r =3

4

a2

=1

23 * a

h = 0.866 a

2 T i



2. Trapezium

Any trapezium can be divided into two triangles. Thus:

A =L * h

2+

l * h

2

=l * h

2* h

A = l m * h

Result

L + l

2

is the length of the rectangle in the diagram.

A2

A1

h

L

l



h

l m

3. Summary

triangle: A =l * h

2

trapezium: A =L + l

2* h

4 Example



4. Example

A =l * h

2

A die with a triangular area has a cross section of 1015

mm2 and a height of 35 mm. Calculate the length of thebase in mm.

find l

given that A = 1015 mm

2

h = 35 mm

solution

l =2 * A

h=

2 * 1015

35

mm2

mm*

l = 58 mm

A

l

h

Ci l



Circular area:

A = aread = diameter

b = arc length

= sector angleα

4

π = 0.785

1.Circle

If we divide a circle into section, we obtain an

approximate parallelogram with

area = base line * height

A =d *

2

π *

2

d=

d 2 *

4

π



1

2

3 4

5

6

7

8

910

11

12

Note

For the calculation of the area or diameter of a circle the

use of tables is recommended.

2. Sector

If we again divide a sector into sectors, we have

b



1

2 3

4

A

d

A =b * r

2

For a full circle of 360o A = d 2 * 0.785

For a sector of α A = d 2 * 0.785360 o

*α

3 Circular ring



3. Circular ring

The difference between two circle areas gives a

circular ring area.

A = A1 – A2

D2 *

4

π = –

d 2 *

4

π

=

4

π * ( D2 – d2 )

A = 0.785 * ( D2 – d2 )

D

4 S



4. Summary

circle A = d 2 * 0.785

sector A = d 2 * 0.785

360 o*

α

A =b * r

2

Circular ring A = A1 – A2 = 0.785 * ( D2

– d 2

)

5 E l



5. Example

For a grinding machine we require a sheet metalcover in the form of a sector with a diameter of 400 mm

and a sector angle of 220o. Calculate the amount of sheet

metal needed in cm2.

find A in cm2

given that d = 400 mm

α = 200o

solution A = d 2 * 0.785

360 o* α

= ( 40 cm )2 * 0.785 220o

360o*

A = 767.56 cm2

C l l ti th l f i id b di



Calculating the volume of rigid bodies

A = area

V = volume

H = height of body

Note

In the case of right bodies, the sides or curved surface are

perpendicular to the parallel base and top area.

1 C b



1. Cubevolume = base area * height

V = A * H

Note

Square base areas can be found in tables.

H

l

2. Prism volume = base area * height

V = A * H

Note

In the case of a prism, the base area can

be of any shape.

H

l

Al

3. Cylinderl b * h i ht



3. Cylindervolume = base area * height

V = A * H

Note

It is easier to find the base number d if

one has the tabular value for the base area A.

H

4. Summary

For calculating the volume of right bodies, we have

volume = base area * height

V = A * H

5. Example



5. Example

The capacity in liters is to be calculated for a

cylindrical water container with a diameter of 350 mm and aheight of 750 mm.

find V in l

given that d = 350 mmH = 750 mm

H

preliminary consideration

volume = base area * height

solution



solution

V = A * H

A = d 2 * 0.785

= 3.52 dm2 * 0.785

A = 9.62 dm2

V = 9.62 dm2 * 7.5 dm = 72.15 dm3

Note

1 dm3 = 1 liter

Calculating the volume of pointed and truncated bodies:



g p

A = area V = volume

H = vertical height of body

1. Cone / Pyramid

In the case of pointed bodies, the slanting lines

converge in a straight line on a point. One prism equals

three pointed bodies in volume if the base area and heightare the same.

V



H

t

V

3V

V = V prism

3

V = base area x height

3

V = A . H

3

Note:

The base area can be of any shape

2 Truncated Cone / Pyramid



2. Truncated Cone / Pyramid

In the case of truncated bodies, the upper part of a

pointed body is cut off parallel to the base. Using the base

and top measurements, we take the mean:

dm = D + d

2

Or Am = A1 + A2

2

Thus we have:

Volume = mean area x height

V = Am x H H

D

d

H

A2

A1m

3 SUMMARY



3. SUMMARY

V pointed = base area x height

3V = A . H

3

V truncated = mean area x height

V = Am x H

4. Example



4. Example

A cone with a diameter of 210mm has a volume of

3056cm3. Calculate the height in cm.

Required: H in cm

Given: V = 3056cm3

d = 210mm

Preliminary Consideration:

Volume of cone = 1/3 volume of cylinder

S l ti V A H



Solution: V = A . H

3

H = 3 . V

A

H = 3 . 3056cm3

346.36cm2

H = 26.47 cm

Calculation of mass



Calculation of mass

m = mass, result of weighting

V = volume

p = density ( pronounce prho )

1. Mass

The Sl unit for mass is 1 kilogram. In practice we also use

derived units for calculation:

1 t = 1000 kg

1 kg = 1000 g

1 g = 1000 mg

Conclusion



The conversion factor from unit to unit is 1000.

1 g 1 kg

watersugar

Masses of comparison

1 Mg = 1 t

1m3

t kg g mg

000 000 000 000

2. DensityWe weigh the mass of a material with a



We weigh the mass of a material with a

volume unit of 1cm3 or 1dm3 or 1m3. We

relate this mass to the volume unit in questionand call the relationship “ DENSITY ”.

g

cm3

kg

dm3

t

m3

m

V

or or = =

Note: These units of mass always belong together

1 2.7

7.85

water

aluminumsteel

ᵨ

3. Summary



The units of mass obtained give the basic equation

Mass = volume x density

m = V x ᵨ

Note : In commerce and industry, the

result of weighing is known as

“weight”.

“Weights” for semi -finished

materials are given in tables.

mass in kg

Weight-force in N

4. Example



Calculate the mass in kg of a rectangular plate with the

Dimensions 220 , 330 and 15 mm.

Required: m in kg

Given : A = 2.2 x 3.3 dm2

s = 0.15 dm

p = 7.85 kg/dm3

Preliminary consideration : there is a relationship between kg and dm3

Solution:



Solution:m = V x ᵨ

V = A x s = 2.2 dm x 3.3 dm x 0.15 dm

= 1.089 dm3

m = 1.089 dm3 x 7.85 kg/dm3 = 8.55 kg

Note : In the case of mass per unit area

(plate weight), it is better to use the

rule of three. For steel, the

statement referring to the unit is :1m2 of 1 mm thickness = 7.85kg

Proof : 1m2 . 1mm = 1dm3, since

100dm2 . 0.01 dm = 1dm3

s

t

Lines and the Cartesian Plane



Lines and the Cartesian Plane

The cartesian plane provides a pallate on which to graph



The cartesian plane provides a pallate on which to graph,

and it is built by taking two number lines and making them

cross at a right angle where both of them are zero.

Then, we have one number line going up

and down, and another number line going from left to right.

Notice that there is an x at the

right end of the horizontal

number line and a y at the top of

the vertical number line. We call

these two number lines axes.Specifically, we call the horizontal

number line the x-axis and the

vertical number line the y-axis.





Each one has its own

number, which we write as a

roman numeral.

That is, quadrants I (1),II (2), III (3) and IV (4).

There are four distinct areas of

the cartesian plane

We name them quadrants

because there are four of them

Notice that in quadrants I and II y is always positive In



Notice that in quadrants I and II, y is always positive. In

quadrants I and IV, x is always positive.

(+,+)

(-,+)

(-,-)

(+,-)

EXAMPLE:What are the coordinates

of the following:



Point M

Point Z

Point D

Point A

M

Z

D A



BASIC TRIGONOMETRICAL FUNCTION

U d it i l t d fi iti t i t i (lit "t i l

TRIGONOMETRY



Under its simplest definition, a trigonometric (lit. "triangle-

measuring") function, is one of the many functions that relateone non-right angle of a right triangle to the ratio of the

lengths of any two sides of the triangle (or vice versa).



Any trigonometric function (f), therefore, always satisfieseither of the following equations:

f(Ɵ) = a / b OR f(a / b) = Ɵ,

where Ɵ is the measure of a certain angle in the triangle, and

a and b are the lengths of two specific sides.



The side lengths (a b c) of a right triangle form a so-



The side lengths (a , b, c) of a right triangle form a so

called Pythagorean triple. A triangle that is not a right triangle

is sometimes called an oblique triangle. Special cases of theright triangle include the isosceles right triangle (middle

figure) and 30-60-90 triangle (right figure).

Applications of Right Triangle

http://mathworld.wolfram.com/PythagoreanTriple.html

http://mathworld.wolfram.com/Triangle.html

http://mathworld.wolfram.com/ObliqueTriangle.html

http://mathworld.wolfram.com/IsoscelesRightTriangle.html

http://mathworld.wolfram.com/30-60-90Triangle.html
















http://mathworld.wolfram.com/Triangle.html






Trigonometric ratios (sine, cosine, and tangent) will be used

in real world applications.

Right triangle, one angle is 90º and the side across from

this angle is called the hypotenuse. The two sides which

form the 90º angle are called the legs of the right triangle.

The legs are defined as either “opposite” or “adjacent” (next

to) the angle A.

We shall call the opposite side "opp," the adjacent side "adj" and

the hypotenuse "hyp."

In the following definitions, sine is called "sin," cosine is called



g

"cos" and tangent is called "tan." The origin of these terms

relates to arcs and tangents to a circle.

i.sin(A) =

ii.cos(A) =

iii.tan(A) =



In a right triangle, there are actually six possible trigonometric ratios, or

Trigonometric Functions



In a right triangle, there are actually six possible trigonometric ratios, or

functions.

A Greek letter (such as theta or phi ) will now be used to represent

the angle.

Notice that the three new ratios at the right are reciprocals of

th ti th l ft



the ratios on the left.

Applying a little algebra shows the connection between these

functions.





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