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 ↓


Trigonometric Equations


A trigonometric equation is a type of equation involving trigonometric functions such as sine, cosine, tangent, and their inverses. These functions relate the angles and sides of triangles, especially right angles, and are fundamental in the study of periodic phenomena, waves, oscillations, and various fields of physics and engineering.

What are trigonometric equations?

A trigonometric equation is an equation that involves one or more trigonometric functions. The goal is often to find all the angles that satisfy the equation. These equations can be simple, such as:

sin(x) = 0.5

or more complex such as:

2cos^2(x) - 3sin(x) + 1 = 0

Basic trigonometric functions

The basic trigonometric functions you will encounter are:

  • sin(x) - the sine of an angle
  • cos(x) - cosine of an angle
  • tan(x) - tangent of an angle
  • csc(x) - the cosecant of an angle, which is 1/sin(x)
  • sec(x) - the secant of an angle, which is 1/cos(x)
  • cot(x) - the cotangent of an angle, which is 1/tan(x)

Understanding the unit circle

The unit circle is fundamental in trigonometry. Understanding the unit circle can help you solve trigonometric equations effectively. The unit circle is a circle with radius 1 and centered at the origin of the coordinate plane.

1, 0 0, 1 0, -1 -1, 0

In this circle:

  • In standard position the angle is measured in the counterclockwise direction from the positive x-axis.
  • At any point on the circle, the x-coordinate gives the cosine of the angle, and the y-coordinate gives the sine of the angle.

Solving trigonometric equations

Solving trigonometric equations involves finding all the angles that satisfy the given equation. Here are the common methods:

Using algebraic techniques

Some trigonometric equations can be solved using algebraic methods or using standard algebra techniques. For example:

2cos^2(x) - cos(x) = 0

You can factor this equation:

cos(x)(2cos(x) - 1) = 0

This leads to two simple equations:

  • cos(x) = 0
  • 2cos(x) - 1 = 0

These can be solved by finding suitable angles satisfying each of these.

Using the identity

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities can make solving trigonometric equations easier. For example:

sin^2(x) + cos^2(x) = 1

This Pythagorean identity can be used to substitute sin^2(x) or cos^2(x) into equations and simplify them.

Graphical methods

Trigonometric equations can also be solved using graphical methods. By plotting the functions involved and looking for the intersection, one can find approximate solutions.

Examples of solving trigonometric equations

Let us look at some examples of solving different types of trigonometric equations.

Example 1: Simple equation

Consider the equation:

sin(x) = 0.5

We need to find all the angles where the sine is equal to 0.5. Looking at the unit circle, we know:

  • x = π/6
  • x = 5π/6

Since sine is periodic and has a time period of 2π, the general solutions are:

  • x = π/6 + 2kπ, where k is an integer
  • x = 5π/6 + 2kπ, where k is an integer

Example 2: Quadratic trigonometric equation

Solve the equation:

2sin^2(x) - sin(x) - 1 = 0

Take sin(x) as a variable, say u. The equation becomes:

2u^2 - u - 1 = 0

Factor this in:

(2u + 1)(u - 1) = 0

Thus, re-substituting u = -1/2 or u = 1 for sin(x), we get:

  • sin(x) = -1/2, which gives the solution x = 7π/6, 11π/6, plus 2kπ
  • sin(x) = 1, which gives the solution x = π/2, plus 2kπ

Example 3: Using identity

Consider:

2sin(x)cos(x) = sin(x)

Use the identity 2sin(x)cos(x) = sin(2x):

sin(2x) = sin(x)

The solutions are as follows:

  • 2x = x + 2kπ gives x = 2kπ
  • 2x = π - x + 2kπ leads to x = π/3 + kπ

Conclusion

Trigonometric equations are an important part of mathematics and appear in many different contexts. By understanding algebraic techniques, trigonometric identities, and the periodic nature of trigonometric functions, one can solve these equations effectively. Visualization using the unit circle and an understanding of the properties of sine, cosine, and other trigonometric functions are the keys to mastering this subject.

Always remember to consider the periodic nature of trigonometric functions and express your solution as a general solution with proper periods.


Grade 11 → 3.2


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