Course Content
Matrices and Transformations
This course introduces students to the fundamental concepts of matrices and their role in transformations. It provides a comprehensive understanding of matrix operations, determinants, and inverses, as well as how matrices are applied to various geometric transformations on the Cartesian plane. Specific Objectives By the end of the topic the learner should be able to: (a) Relate image and object under a given transformation on the Cartesian Plane; (b) Determine the matrix of a transformation; (c) Perform successive transformations; (d) Determine and identify a single matrix for successive transformation; (e) Relate identity matrix and transformation; (f) Determine the inverse of a transformation; (g) Establish and use the relationship between area scale factor and determinant of a matrix; (h) Determine shear and stretch transformations; (i) Define and distinguish isometric and non-isometric transformation; (j) Apply transformation to real life situations. Content (a) Transformation on the Cartesian plane (b) Identification of transformation matrix (c) Successive transformations (d) Single matrix of transformation for successive transformations (e) Identity matrix and transformation (f) Inverse of a transformations (g) Area scale factor and determinant of a matrix (h) Shear and stretch (include their matrices) (i) Isometric and non-isometric transformations (j) Application of transformation to real life situations.
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Introduction to Matrices
Introduction to Matrices
00:10:00
Statistics II
This course introduces students to the fundamental concepts of statistics, focusing on measures of central tendency , cumulative frequency tables and ogives, and measures of dispersion. By the end of this module, students will have a strong understanding of how to organize, analyze, and interpret data using various statistical methods. Specific Objectives By the end of the topic the learner should be able to: (a) State the measures of central t e n d e n c y; (b) Calculate the mean using the assumed mean method; (c) Make cumulative frequency table, (d) Estimate the median and the quartiles b y - Calculation and - Using ogive; (e) Define and calculate the measures of dispersion: range, quartiles,interquartile range, quartile deviation, variance and standard deviation (f) Interpret measures of dispersion Content (a) Mean from assumed mean: (b) Cumulative frequency table (c) Ogive (d) Meadian (e) Quartiles (f) Range (g) Interquartile range (h) Quartile deviation (i) Variance (j) Standard deviation These statistical measures are called measures of central tendency and they are mean, mode and median
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Statistics II
Past KCSE Questions on the Lesson
THREE-DIMENSIONAL GEOMETRY
Specific Objectives By the end of the topic the learner should be able to: (a) State the geometric properties of common solids; (b) Identify projection of a line onto a plane; (c) Identify skew lines; (d) Calculate the length between two points in three dimensional geometry; (e) Identify and calculate the angle between (i) Two lines; (ii) A line and a plane; (ii) Two planes. Content (a) Geometrical properties of common solids (b) Skew lines and projection of a line onto a plane (c) Length of a line in 3-dimensional geometry (d) The angle between i) A line and a line ii) A line a plane iii) A plane and a plane iv) Angles between skewlines.
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3-D Geometry
Probability II
This course delves into advanced probability concepts, focusing on both theoretical and experimental probability. It introduces probability spaces, combined events, and probability laws while incorporating visual tools like tree diagrams for better understanding. Key Topics Covered: 1. Probability: - The measure of how likely an event is to occur. - Expressed as a fraction, decimal, or percentage. - Example: Rolling a die and getting a 3 (P = 1/6). 2. Experimental Probability: - Probability based on actual experiments or observations. - Example: Flipping a coin 100 times and recording how many heads appear. 3. Range of Probability Measure: - Probability values are always between 0 (impossible event) and 1 (certain event). 4. Probability Space: - A set of all possible outcomes of an experiment. - Includes: - Sample space (S): All possible outcomes. - Events (E): A subset of the sample space. - Example: Rolling a six-sided die → Sample space: {1, 2, 3, 4, 5, 6}. 5. Theoretical Probability: - Probability determined using logic rather than experiments. - Example: The probability of drawing a red card from a standard deck is 26/52 = 1/2. 6. Discrete and Continuous Probability: - Discrete Probability: Deals with countable outcomes (e.g., rolling a die). - Continuous Probability: Deals with uncountable outcomes over an interval (e.g., height of students). 7. Combined Events: - Mutually Exclusive Events: Events that cannot happen at the same time. - Example: Getting heads and tails in a single coin toss. - Independent Events: Events where one does not affect the probability of the other. - Example: Rolling two dice. 8. Laws of Probability: - Addition Law: Used for mutually exclusive events. - Multiplication Law: Used for independent events. 9. Tree Diagrams: - A visual representation of probabilities in multi-step experiments. - Used to calculate the probability of sequences of events. - Example: Finding the probability of getting two heads in a row when flipping a coin. Learning Outcomes: By the end of this course, students will be able to: - Differentiate between experimental and theoretical probability. - Understand probability spaces and how to define sample spaces. - Apply probability laws to mutually exclusive and independent events. - Construct and analyze tree diagrams for multi-stage events. - Use probability in real-world applications like statistics, risk assessment, and decision-making. Why This Course Matters: Probability is essential in daily decision-making, finance, health sciences, and artificial intelligence. Mastering these concepts enhances logical reasoning and prepares students for advanced mathematical studies and KCSE exams.
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Probability
TRIGONOMETRY III
This topic explores trigonometric ratios, identities, graphing, and solving trigonometric equations. Students will learn how to analyze trigonometric functions and their key properties, such as amplitude, period, and phase angle. Key Learning Objectives By the end of this topic, students will be able to: 1. Recall and define trigonometric ratios (sine, cosine, and tangent). 2. Derive and apply the fundamental identity: sin2x+cos2x = 1 3. Plot and interpret graphs of trigonometric functions. 4. Solve simple trigonometric equations both analytically and graphically. 5. Determine amplitude, period, wavelength, and phase angle from trigonometric graphs. Content Overview (a) Trigonometric ratios (b) Deriving the relation sin2x+cos2x =1 (c) Graphs of trigonometric functions of the form y = sin x y = cos x, y = tan x y = a sin x, y = a cos x, y = a tan x y = a sin bx, y = a cos bx y = a tan bx y = a sin(bx ± 9) y = a cos(bx ± 9) y = a tan(bx ± 9) (d) Simple trigonometric equation (e) Amplitude, period, wavelength and phase angle of trigonometric functions. Why This Topic Matters Trigonometry is fundamental in physics, engineering, and real-world applications such as sound waves, light waves, and mechanical vibrations. Understanding trigonometric functions and their properties is essential for solving complex problems in science and technology.
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Trigonometry III
Past KCSE Questions
Vectors II
Alright, let’s break down Vectors II like it’s your ultimate cheat sheet for conquering 2D and 3D space. We’re diving into the nitty-gritty of representing vectors, doing the math with them, and applying these skills to geometry like a boss.
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Vectors II
Form 4 Mathematics Online Tuition
About Lesson

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of a right-angled triangle. They are fundamental in solving geometric and real-world problems involving angles.

 

 

1. Defining Trigonometric Ratios

Consider a right-angled triangle with:

  • Hypotenuse (r): The longest side opposite the right angle.
  • Opposite (O): The side opposite the given angle θtheta.
  • Adjacent (A): The side next to the given angle θtheta.

 

 

Primary Trigonometric Ratios

  • Sine (sin θ):
    sin(θ) = Opposite / Hypotenuse = O / r

  • Cosine (cos θ):
    cos(θ) = Adjacent / Hypotenuse = A / r

  • Tangent (tan θ):
    tan(θ) = Opposite / Adjacent = O / A

 

 

Reciprocal Trigonometric Ratios

  • Cosecant (csc θ):
    csc(θ) = 1 / sin(θ) = r / O

  • Secant (sec θ):
    sec(θ) = 1 / cos(θ) = r / A

  • Cotangent (cot θ):
    cot(θ) = 1 / tan(θ) = A / O

 

 

2. Trigonometric Ratios for Special Angles

Angle θtheta sin⁡θsin theta cos⁡θcos theta tan⁡θtan theta
0° 0 1 0
30° 12 1/2 32 + √3/2 13 + 1/√3
45° 22 + √2/2 22 + (√2/2) 1
60° 32 + (√3/2) 12 1/2 3√3
90° 1 0 Undefined

 

 

 

3. Applications of Trigonometric Ratios

  • Navigation & Aviation: Used to calculate angles and distances.
  • Architecture & Engineering: Designing buildings and bridges.
  • Physics & Astronomy: Measuring forces and celestial distances.

 

Waves

Amplitude
This is the maximum displacement of the wave above or below the x axis.
Period
The interval after which the wave repeats itself

 

 

Transformations of waves

The graphs of y = sin x and y = 3 sin x can be drawn on the same axis. The table below gives the corresponding values of sin x and 3 sin x for

 

Angle (°) sin(x) 3 sin(x)
0 0 0
30 0.50 1.50
60 0.87 2.61
90 1.00 3.00
120 0.87 2.61
150 0.50 1.50
180 0 0
210 -0.50 -1.50
240 -0.87 -2.61
270 -0.50 -1.50
300 -0.87 -2.61
330 -0.50 -1.50
360 0 0
390 0.50 1.50
420 0.87 2.61
450 1.00 3.00
480 0.87 2.61
510 0.50 1.50
540 0 0
570 -0.50 -1.50
600 -0.87 -2.61
630 -1.00 -3.00
660 -0.87 -2.61
690 -0.50 -1.50
720 0 0

 

 

The wave of y = 3 sin x can be obtained directly from the graph of y = sin x by applying a stretch scale
factor 3 , x axis invariant .

 

 

Note;
• The amplitude of y= 3sinx is y =3 which is three times that of y = sin x which is y =1.
• The period of the both the graphs is the same that is or 2

 

 

 

 

Example
Draw the waves y = cos x and y = cos . We obtain y = cos from the graph y = cos x by applying a stretch of factor 2 with y axis invariant.

Note;
• The amplitude of the two waves are the same.
• The period of y = cos is that is, twice the period of y = cos x

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