Course Aim

To build a strong foundation in mathematical concepts essential for problem-solving, navigation, and technical operations in maritime professions. 

▶️ Understanding Exponents and Their Operations 🕗8:41 Minutes [Click Here]

▶️ Order of Arithmetic Operations: PEMDAS 🕘4:21 Minutes [Click Here]

▶️ Least Common Multiple 🕘5:39 Minutes [Click Here]

▶️ Greatest Common Factor 🕘4:46 Minutes [Click Here]

▶️ Introduction to Algebra: Using Variables 🕘4:03 Minutes [Click Here]

▶️ Algebraic Properties 🕘5:15 Minutes [Click Here]

▶️ Algebraic Equations and Their Solutions 🕘4:51 Minutes [Click Here] 

▶️ Algebraic Equations With Variables on Both Sides 🕘6:45 Minutes [Click Here]

▶️ Solving Algebraic Equations with Roots and Exponents 🕘5:45 Minutes [Click Here]

Algebra is a convenient system of notation in which letters or symbols are used to represent quantities, instead of words.

Example

Suppose a box contained 25 bolts, 34 nuts and 47 washers. To the contents 15 bolts and 18 washers were added but 14 nuts were removed.

This could be set down as follows:

       25b + 34n + 47w

add    15b – 14n + 18w

-----------------------

result  40b + 20n + 65w

Example

The area of a rectangle is found by its length multiplied by its width.

If the area is A, the length L and the width W, then
A = L × W.

If the length and width are subtracted then the answer is L – W, if they are added L + W and if they are divided L ÷ W or L / W.

Note: the ‘×’ symbol is often ignored so L × W becomes LW (or L · W)

When a quantity, y, is multiplied by a number, for example 3, the answer is written 3y. If a quantity appears by itself, for example k, it means one times k.

If a combination of letters and numbers are multiplied together the order does not matter so d4e is the same as ed4 and so forth. However, it is usual to write the number first and the letters alphabetical afterwards, i.e. 4de.

An expression which consists of two terms is called a binomial expression, and one consisting of three terms is called a trinomial expression.

Addition of terms

Example
x + 3x = 4x but c + d stays as c + d.

Example
Add 2a + 7b + 3c and 6c + 5a − 2b

    2a + 7b + 3c  

Add 5a − 2b + 6c  

-----------------  

    7a + 5b + 9c

Example
Add 3x − 5y + 6z and 4x + 2y − 8z

    3x − 5y + 6z  

Add 4x + 2y − 8z  

-----------------  

    7x − 3y − 2z

Subtraction of terms

Example

From 8x subtract 5x

8x − 5x = 3x

From 8x subtract −5x
8x − (−5x) = 8x + 5x = 13x

From −8x subtract 5x
−8x − 5x = −13x

From −8x subtract −5x
−8x − (−5x) = −8x + 5x = −3x

Example

Subtract 6c − 3b + 4a − 7d from 5b + 3d + 2a − 9c

   4a − 3b + 6c − 7d   subtract

− (2a + 5b − 9c + 3d)

---------------------------

   2a − 8b + 15c − 10d

However, 2abc + 5bac = 7abc as each term has the same set of letters although the order differs.

m²n and mn² are different, as the first term is m × m × n and the second is m × n × n.

Solving Higher-Degree Polynomials by Synthetic Division and the Rational Roots Test

🕘9:21 Minutes [Click Here]

▶️ Useful wesite link to learn Algebra: [Click Here]

▶️ Graphing in Algebra: Ordered Pairs and the Coordinate Plane 🕘6:55 Minutes [Click Here] 

Solving Systems of Two Equations and Two Unknowns: Graphing, Substitution, and Elimination 🕘10:20 Minutes

[Click Here]

An equation is an expression consisting of two ‘sides’, one being equal in value to the other. A simple equation contains one hidden value of the first order (e.g. x, and not x², or x³, etc.) which is usually referred to as the unknown and to solve the equation means to find the value of the unknown.

Since one side of the equation is always equal to the other side, it follows that if, in the process of solving the equation, a change is made to one side of the equation the same change must be made to the other side in order to maintain equality.

The most important thing to remember is that the equality of the equation must be maintained.

This means if the values on the left-hand side of the equals sign are doubled then the values on the right-hand side must also be doubled.

Any mathematical operation can be used, so long as the ‘balance’ of the equation is maintained.

So equality will be maintained if
(i) the same quantity is added to both sides,
(ii) the same quantity is subtracted from both sides,
(iii) every term on both sides is multiplied by the same value,
(iv) every term on both sides is divided by the same value,
(v) both sides are raised to the same power,
(vi) the same root is taken of both sides.

Methods of approach
(i) eliminate fractions by multiplying all the terms by the LCM of the denominators,
(ii) remove brackets, following the rules of Algebra.

(iii) place all the terms that involve the unknown on one side of the equation, and all other terms on the other side of the equation,
(iv) collect and summarise terms on each side,
(v) find the value of the unknown.

This is a rough guide only.

Example
Solve the equation: 6x = 24
Dividing both sides of the equation by 6 gives: 6x ÷ 6 = 24 ÷ 6
Cancelling gives: x = 4

Every equation should be checked by substituting the answer into the original equation.
In this case, 6 × 4 = 24, so the answer is correct.

Example
Solve the equation: (2x / 3) = 12
Multiplying both sides by 3 gives: (2x / 3) × 3 = 12 × 3
Cancelling gives: 2x = 36
Dividing both sides by 2 gives: 2x ÷ 2 = 36 ÷ 2
Cancelling gives: x = 18
Check: (2 × 18) / 3 = 36 / 3 = 12. Therefore, answer is correct.

Example
Solve the equation: x − 8 = 3
Adding 8 to both sides gives: x − 8 + 8 = 3 + 8
So, x = 11
Check: 11 − 8 = 3, so correct.

Example
Solve the equation: x + 5 = 9

Subtracting 5 from both sides gives:
x + 5 − 5 = 9 − 5
So, x = 4
Check: 4 + 5 = 9, so correct.

Example
Solve the equation: 4x + 10 = 18
Subtracting 10 from both sides gives: 4x = 8
Dividing both sides by 4 gives: x = 2
Check: 4 × 2 + 10 = 8 + 10 = 18, so correct.

Example
Solve the equation: 5x + 6 = 3x + 12

In questions like this, all the terms involving the unknown quantity (in this case, x) must be grouped on one side of the equals sign with everything else on the other side.

When this is done, changing from one side of an equals sign to the other must be accompanied by a change of sign.

5x + 6 = 3x + 12
5x − 3x = 12 − 6
2x = 6, so x = 6 ÷ 2 = 3

Check: (5 × 3) + 6 = 21, and (3 × 3) + 12 = 21, so correct.

Example
Solve the equation: 6 − 3x = 4x − 20

In order to keep the x terms positive, move the x’s to the right-hand side of the equals sign, and everything else to the left-hand side.

6 − 3x = 4x − 22
6 + 22 = 4x + 3x
28 = 7x
x = 28 ÷ 7 = 4

Check:
6 − 3 × 4 = 6 − 12 = −6
4 × 4 − 22 = 16 − 22 = −6, so correct.

Example
Solve the equation: 5(x − 2) = 25

Removing the bracket gives:
5x − 10 = 25

Rearranging gives:
5x = 25 + 10
5x = 35
x = 35 ÷ 5 = 7

Check:
5(7 − 2) = 5 × 5 = 25, so correct.

Alternatively, divide both sides by 5, then add 2 to both sides.

Example
Solve the equation:
3(2x − 6) + 2(x + 3) = 4(x − 5)

Removing the brackets gives:
6x − 18 + 2x + 6 = 4x − 20

Collecting like terms gives:
6x + 2x − 4x = −20 + 18 − 6

Simplifying gives:
4x = −8

So x = −8 ÷ 4 = −2

Check:
3(−4 − 6) + 2(−2 + 3) = 3 × −10 + 2 × 1 = −30 + 2 = −28
4(−2 − 5) = 4 × −7 = −28, so correct.

▶️ Basic Euclidean Geometry: Points, Lines, and Planes 🕟4:18 Minutes [Click Here]

 ▶️ Types of Angles and Angle Relationships 🕟7:23 Minutes [Click Here]

▶️ Types of Triangles in Euclidean Geometry 🕟5:24 Minutes [Click Here]

▶️ Proving Triangle Congruence and Similarity 🕟5:07 Minutes [Click Here]

▶️ Special Lines in Triangles: Bisectors, Medians, and Altitudes 🕟6:19 Minutes [Click Here]

▶️ The Triangle Midsegment Theorem 🕟3:07 Minutes [Click Here]

▶️ The Pythagorean Theorem 🕟5:01 Minutes [Click Here]

▶️ Types of Quadrilaterals and Other Polygons 🕟6:16 Minutes [Click Here]

▶️ Calculating the Perimeter of Polygons 🕟5:07 Minutes [Click Here]

▶️ Circles: Radius, Diameter, Chords, Circumference, and Sectors 🕟4:57 Minutes [Click Here]

▶️ Calculating the Area of Shapes 🕟6:38 Minutes [Click Here]

▶️ Proving the Pythagorean Theorem 🕟3:33 Minutes [Click Here]

▶️ Three-Dimensional Shapes Part 1: Types, Calculating Surface Area 🕟7:31 Minutes [Click Here]

▶️ Three-Dimensional Shapes Part 2: Calculating Volume 🕟6:16 Minutes [Click Here]

Unit 1: Functions

▶️ Back to Algebra: What are Functions? 🕘 7:22 Minutes [Click Here]

▶️ Manipulating Functions Algebraically and Evaluating Composite Functions 🕘5:28 Minutes [Click Here]

▶️ Graphing Algebraic Functions: Domain and Range, Maxima and Minima 🕘5:53 Minutes [Click Here]

▶️ Transforming Algebraic Functions: Shifting, Stretching, and Reflecting 🕘7:51 Minutes [Click Here]

▶️ Continuous, Discontinuous, and Piecewise Functions 🕘5:17 Minutes [Click Here]

▶️ Inverse Functions 🕘6:28 Minutes [Click Here]

▶️ The Distance Formula: Finding the Distance Between Two Points 🕔3:37 Minutes [Click Here]

Unit 2: Conic Sections Part 1: Circles

▶️ Graphing Conic Sections Part 1: Circles 🕘 7:45 Minutes [Click Here]

▶️ Graphing Conic Sections Part 2: Ellipses 🕘6:32 Minutes [Click Here]

▶️ Graphing Conic Sections Part 3: Parabolas in Standard Form 🕘 8:29 Minutes [Click Here]

▶️ Graphing Conic Sections Part 4: Hyperbolas 🕘 5:56 Minutes [Click Here]

▶️ Graphing Higher-Degree Polynomials 🕘9:14 Minutes [Click Here]

▶️ Graphing Rational Functions and Their Asymptotes 🕘7:24 Minutes [Click Here]

Unit 3: Logarithms Part 1

▶️ Logarithms Part 1: Evaluation of Logs and Graphing Logarithmic Functions 🕔8:09 Minutes [Click Here]

▶️ Logarithms Part 2: Base Ten Logs, Natural Logs, and the Change-Of-Base Property 🕞6:25 Minutes [Click Here]

▶️ Logarithms Part 3: Properties of Logs, Expanding Logarithmic Expressions 🕗7:05 Minutes [Click Here]

▶️ Solving Exponential and Logarithmic Equations 🕘7:07 Minutes [Click Here]

Unit 4: Set Theory: Types of Sets, Unions and Intersections

▶️ Set Theory: Types of Sets, Unions and Intersections 🕘6:21 Minutes [Click Here]

Unit 5: Sequences

▶️ Sequences, Factorials, and Summation Notation 🕙11.11 Minutes [Click Here]

Unit 6: Probability

▶️ Theoretical Probability, Permutations and Combinations 🕘15:51 Minutes [Click Here]

Unit 1: Introduction to Trigonometry

▶️ Introduction to Trigonometry: Angles and Radians 🕘6:26 Minutes [Click Here]

An angle is the corner of two joining lines and the magnitude of an angle is measured in either degrees or radians.

Measurement of Angles

The most common unit of angle measurement that is known is probably the degree. There are 360° in a circle and so a degree is one three-hundred-and-sixtieth part of a circle.

The symbol for a degree is ° and so there are 360° in a circle.

Figure 7.1 shows a quarter of a circle which is 90° and is termed a right-angle, an angle which is less than 90° (Figure 7.2) is called an acute angle, greater than 90° but less than 180° (Figure 7.3) is an obtuse angle, and greater [than] 180° (Figure 7.4) is a reflex angle.

One sixtieth part of a degree is termed one minute and the sixtieth part of a minute is one second.

Symbols are used to represent degrees, minutes and seconds.

60 s = 60″ = 1 min;
60 min = 60′ = 1°;
360° = 1 circle.

An angle of 35 degrees 23 minutes and 15 seconds is written as 35° 23′ 15″.

Notation depends on the circumstance: some situations require angles to be given in decimal form accurate to, say, 3 decimal places; others require answers correct to the nearest tenth of one minute, for example 35° 25.7′; and others may require an accuracy correct to the nearest second, for example 35° 23′ 15″.

Unit 2: Trigonometric Functions

▶️ Trigonometric Functions: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent 🕘7:17 Minutes [Click Here]

▶️ The Easiest Way to Memorize the Trigonometric Unit Circle 🕘9:47 Minutes [Click Here]

▶️ Basic Trigonometric Identities: Pythagorean Identities and Cofunction Identities 🕘5:24 Minutes [Click Here]

▶️ Graphing Trigonometric Functions 🕘11:39 Minutes [Click Here]

▶️ Inverse Trigonometric Functions 🕘6:53 Minutes [Click Here]

▶️ Verifying Trigonometric Identities 🕘9:13 Minutes [Click Here]

▶️ Formulas for Trigonometric Functions: Sum/Difference, Double/Half-Angle 🕘9:28 Minutes [Click Here]

▶️ Solving Trigonometric Equations 🕘8:27 Minutes [Click Here]

Unit 3: The Law of Sines

▶️ The Law of Sines 🕘5:13 Minutes [Click Here]

Unit 4: The Law of Cosines

▶️ The Law of Cosines 🕘4:37 Minutes [Click Here]

Unit 5: Polar Coordinates

▶️ Polar Coordinates and Graphing Polar Equations 🕘10:45 Minutes [Click Here]

Unit 6: Parametric Equations

▶️ Parametric Equations 🕘4:35 Minutes [Click Here]

Unit 1: Introduction to Calculus

▶️ Introduction to Calculus: The Greeks, Newton, and Leibniz 🕘8:39 Minutes [Click Here]

Unit 2: Differentiation

▶️ Understanding Differentiation Part 1: The Slope of a Tangent Line 🕘5:28 Minutes [Click Here]

▶️ Understanding Differentiation Part 2: Rates of Change 🕘5:30 Minutes [Click Here]

Unit 3: Limits and Limit Laws

▶️ Limits and Limit Laws in Calculus 🕘12:48 Minutes [Click Here]

Unit 4: Derivative

▶️ What is a Derivative? Deriving the Power Rule 🕘10:04 Minutes [Click Here]

▶️ Derivatives of Polynomial Functions: Power Rule, Product Rule, and Quotient Rule 🕘11:52 Minutes [Click Here]

▶️ Derivatives of Trigonometric Functions 🕘7:56 Minutes [Click Here]

▶️ Derivatives of Composite Functions: The Chain Rule 🕘12:28 Minutes [Click Here]

▶️ Derivatives of Logarithmic and Exponential Functions 🕘8:40 Minutes [Click Here]

Unit 5: Implicit Differentiation

▶️ Implicit Differentiation 🕘11:44 Minutes [Click Here]

▶️ Higher Derivatives and Their Applications 🕘7:28 Minutes [Click Here]

▶️ Related Rates in Calculus 🕘8:52 Minutes [Click Here]

▶️ Finding Local Maxima and Minima by Differentiation 🕘6:16 Minutes [Click Here]

▶️ Graphing Functions and Their Derivatives 🕘13:05 Minutes [Click Here]

▶️ Optimization Problems in Calculus 🕘10:54 Minutes [Click Here]

Unit 6: Integration

▶️ What is Integration? Finding the Area Under a Curve 🕘8:17 Minutes [Click Here]

▶️ The Fundamental Theorem of Calculus: Redefining Integration 🕘9:37 Minutes [Click Here]

▶️ Properties of Integrals and Evaluating Definite Integrals 🕘9:47 Minutes [Click Here]

▶️ Evaluating Indefinite Integrals 🕘10:43 Minutes [Click Here]

▶️ Evaluating Integrals With Trigonometric Functions 🕘7:31 Minutes [Click Here]