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 ↓


Cosine Graph


In trigonometry, the cosine function is a fundamental function used to relate the angles of a triangle to the lengths of its sides. The cosine graph is a visual representation of this function, showing how it changes as the angle increases. This article will explain in detail the cosine graph, its properties, and its relation to real-world situations.

Introduction to the cosine function

The cosine function is a recurring function that appears abundantly in mathematics, physics, engineering, and many other fields. In its mathematical form, the cosine function is represented as:

f(x) = cos(x)

This equation tells us that for any angle x (measured in radians), the function returns the cosine of that angle. The cosine function arises from the unit circle, a circle with radius one and centered at the origin of the coordinate plane. When you draw a point on the circle that makes an angle x with the positive x-axis, the x-coordinate of this point is equal to cos(x).

Basic properties of the cosine graph

The cosine graph exhibits several key properties that make it unique and useful:

  • Periodicity: The cosine function repeats every 2π radians. This means that cos(x) = cos(x + 2πk) for any integer k.
  • Amplitude: The maximum value of the cosine function is 1, and the minimum value is -1, that is, its amplitude is 1.
  • Symmetry: The cosine graph is an even function, which means it is symmetric about the y-axis. This means cos(-x) = cos(x).
  • Range: The value of the cosine function lies between -1 to 1.
  • X-intercepts: These occur at (π/2 + nπ, 0) for any integer n.
  • Y-intercept: The graph intersects the y-axis at (0, 1).

Creating a cosine graph

To better understand the cosine graph, let's plot it over a full cycle from 0 to 2π.

0 -1 1 X 1 -1 cos(x)

In the above graph:

  • The x-axis shows the angle x in radians, from 0 to 2π.
  • The y-axis represents the value of cos(x).
  • The graph starts at (0, 1), reaches (π/2, 0), falls to (π, -1), rises again to (3π/2, 0), and ends the cycle at (2π, 1).

Transform of cosine graph

Like other trigonometric functions, the cosine graph can be transformed in various ways, including shift, stretch, and reflection. We will discuss some common transformations.

Vertical stretch and compression

Vertical stretching or compression changes the amplitude of the cosine graph. The function can be written as:

f(x) = A cos(x)

where A is a constant. If |A| > 1, the graph is stretched vertically, increasing the amplitude. If |A| < 1, the graph is compressed.

For example, consider:

f(x) = 2 * cos(x)

Here, the amplitude is increased to 2. This stretches the graph such that it now ranges from -2 to 2.

Horizontal shift

Horizontal shift involves moving the graph left or right along the x-axis. Horizontal shift is represented as:

f(x) = cos(x - C)

where C is the shift. If C > 0, the graph shifts to the right by C units, while if C < 0, it shifts to the left.

For example, consider:

f(x) = cos(x - π/4)

In this graph, the cosine wave is shifted to the right by π/4.

Vertical shift

Vertical shifts move the graph up or down, modifying its baseline. This is expressed as:

f(x) = cos(x) + D

Where D is the vertical displacement. If D > 0, the graph shifts upward, and if D < 0, it shifts downward.

For example, consider:

f(x) = cos(x) + 1

Here, the entire graph is shifted 1 unit upward.

Phase shift

Phase change is the combination of horizontal and vertical changes and can be expressed as:

f(x) = cos(Bx - C)

This shifts the graph horizontally by C/B units. The constant B affects the period and phase of the cosine wave.

For example:

f(x) = cos(2x - π/2)

In this example, the function experiences a phase shift to the right by π/4.

Real-world applications of the cosine function

The cosine function has many practical applications in various fields. Its periodic nature makes it ideal for modeling cyclic phenomena. Below are some examples:

Sound waves

Sound waves act in a periodic manner, and the cosine function is often used to model simple harmonic motions, such as pure tones.

Light waves

Just like sound waves, light waves also exhibit periodic behavior. Cosine functions can describe oscillations in the electromagnetic field associated with light waves.

Circular motion

When an object moves in a circle at constant speed, its projection on the x and y axes is modeled using cosine and sine functions respectively. This is evident in circular motion analysis in physics.

Conclusion

The cosine graph is a powerful tool in mathematics and science, helping to depict and analyze periodic behaviors. Through understanding its properties and transformations, one can apply the cosine function to a multitude of real-life phenomena and mathematical problems. By mastering these concepts, you establish a solid foundation for further exploring and understanding the beauty and applicability of trigonometric functions.


Grade 11 → 3.3.2


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Cosine Graph

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