Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Trigonometry → Graphs of Trigonometric Functions ↓


Sine Graph


In trigonometry, the sine graph is an essential component that helps us understand periodic functions, which are functions that repeat themselves at a certain interval. The sine graph is the visual representation of the sine function which is one of the fundamental trigonometric functions related to a right triangle. This function calculates the ratio of the length of the side opposite a given angle to the hypotenuse. The sine function is represented as sin(θ).

Understanding the basic form of the sine function

The sine function is defined mathematically as:

y = sin(x)

This function takes an angle x as input and outputs the sine of that angle, represented as y. The angle x is usually measured in radians, although it can also be measured in degrees. For simplicity, we often use the unit circle (a circle with radius 1) to represent the sine function, which makes the hypotenuse always equal to 1.

Periodicity and amplitude

The graph of the sine function forms a wave-like pattern known as a sine wave. The two primary characteristics of this wave are its amplitude and its period:

  • Amplitude: The amplitude of a sine wave is the maximum value the function reaches. For the basic sine function y = sin(x), the amplitude is 1. This is because the sine of an angle can never be greater than 1 or less than -1 on the unit circle.
  • Period: The period of a sine function is the interval during which the function completes one full cycle and begins to repeat. For y = sin(x), the period is 2π radians (or 360 degrees).

Looking at a basic sine graph

When plotted, the basic sine graph looks like a smooth wave that oscillates up and down the x-axis. It starts at the origin, goes up to 1 at π/2 radians (90 degrees), returns to 0 at π radians (180 degrees), descends to -1 at 3π/2 radians (270 degrees), and returns back to 0 at 2π radians (360 degrees).

y = sin(x) 0 π 2π -π -1 1 1 -1

Transformation of sine graph

A sine graph can be transformed through transformations that change its appearance. These transformations include amplitude, period, horizontal shift, and vertical shift.

1. Changing dimensions

The amplitude of a sine wave can be changed by multiplying the sine function by a constant. This modifies the height of the wave without affecting its period.

y = a * sin(x)

The value of a is the amplitude. If a is greater than 1, the wave will stretch vertically. If a is between 0 and 1, it will compress.

y = sin(x) y = 0.5 * sin(x)

2. Changing times

The period of the sine function can be changed by multiplying the variable x by a constant. This changes the frequency of the waves.

y = sin(b * x)

The period is calculated as 2π / |b|. The larger the value of b, the smaller the period, resulting in more waves within the same distance.

y = sin(x) y = sin(2x)

3. Horizontal shift

You can shift the sine wave left or right by adding or subtracting a constant inside the function.

y = sin(x - c)

If c is positive then it shifts the graph to the right by c units and if c is negative then it shifts the graph to the left.

y = sin(x) y = sin(x - π/4)

4. Vertical shift

The sine graph can also be shifted up or down by adding or subtracting a constant from the function.

y = sin(x) + d

If d is positive then it moves the graph up by d units, and if d is negative then it moves the graph down.

y = sin(x) y = sin(x) - 0.5

Applications of sine graphs

The sine graph and its transformations are not only academically interesting but are also important for understanding various practical applications. These applications span many fields, including physics, engineering, and even music.

1. Sound waves

Many sounds are made up of sine waves, the simplest of which are pure tones. When several sine waves combine, more complex waves are produced, creating the different sounds we perceive in our environment.

2. Light waves

Light waves can also be analyzed using the sine function. Different colors of light are effectively components of different wavelengths represented by sine waves.

3. Pendulum movement

The motion of a simple pendulum can be modeled using the sine function, which helps predict the position of the pendulum at any point in its swinging motion.

These few examples demonstrate the ubiquity of the sine wave in describing periodic and oscillatory behavior.

Conclusion

Understanding sine graphs is fundamental to recognizing patterns in cyclic phenomena in both nature and technology. With a foundation in unit circle definitions and transformations, sine graphs enable students and professionals to model and predict behavior in a variety of scientific and engineering contexts. As you practice interpreting and transforming sine waves, you will develop intuition about periodic functions and how they appear in the real world.


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Sine Graph

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