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 ↓


General Solutions in Trigonometric Equations


In trigonometry, we often deal with equations involving trigonometric functions such as sine, cosine, and tangent. These equations may have multiple solutions, especially within the various cycles of the periodic nature of the trigonometric functions. Identifying these solutions and expressing them in general form is important for solving trigonometric equations.

Understanding trigonometric functions

Before discussing general solutions, let's recall some properties of trigonometric functions:

  • Sine function: sin(x) is a periodic function with period 2π.
  • The cosine function: cos(x) also has a period of 2π.
  • The tangent function: tan(x) is periodic with period π.

These functions repeat their values in their respective periods. For example, if sin(x) = a then sin(x + 2πk) = a for any integer k.

X Y π 2π

What are the common solutions?

A general solution in trigonometry refers to a formula that expresses all possible solutions of a trigonometric equation. Given the periodic nature of trigonometric functions, there are infinitely many solutions at regular intervals. These regular intervals are defined by the period of the function.

The general solution is usually given as:

 x = x₀ + nT

Where:

  • x₀ is a specific solution within the first cycle (usually between 0 and the period).
  • n is any integer that represents how many cycles you move forward or backward.
  • T is the period of the function (for example, 2π for sine and cosine).

Example of finding a general solution

Let's figure out how to find the general solution to the basic trigonometric equation:

Example 1: Solve for x in sin(x) = 0.5.

First, find x within the first cycle (0 to 2π):

  1. We know that sin(π/6) = 0.5.
  2. Given the property of the sine function, another solution in the same cycle is x = π - π/6 = 5π/6.

To write the general solution:

 x = π/6 + 2πn

And

 x = 5π/6 + 2πn

for any integer n.

Explanation:

The solutions π/6 and 5π/6 serve as the basis within the period of one complete trigonometric cycle of 2π. Since the sine function is periodic, each complete cycle results in the function output being repeated. Thus, the general solutions take into account moving through the cycles with the term 2πn.

π/6 5π/6

Example 2: Solve for x in cos(x) = -0.5.

First, find x within the first cycle (0 to 2π):

  1. We know that cos(2π/3) = -0.5.
  2. Given the properties of the cosine function, another solution in the same cycle is x = 4π/3.

To express the general solution, we have:

 x = 2π/3 + 2πn

And

 x = 4π/3 + 2πn

for any integer n.

Explanation:

By identifying two specific solutions within the 0 to 2π interval, we can understand the periodic nature of the cosine function. The two solutions indicate a symmetry about π axis (because cosine is negative when the angle is in the second or third quadrant). 2π added to these values creates other feasible points because the cosine values repeat every full cycle.

2π/3 4π/3

Example 3: Solving the tan equation

Now, consider an equation related to the tangent function:

tan(x) = 1

Since the tangent function has a period of π, solving tan(x) = 1 in the interval from 0 to π gives:

  1. Principal value x = π/4 since tan(π/4) = 1.

For the general solution:

 x = π/4 + πn

for any integer n.

Comment:

The general solution for tangent may seem simple because the zeros of the tangent occur every π radians. Therefore, adding the period π gives the same tan value.

π/4

Importance of the general solution

Mastering the general solutions provides a deeper understanding of trigonometric equation behavior across the entire set of real numbers, rather than restricting the solutions to a finite interval. These solutions allow us to predict and explain every occurrence of an angle that fits the equation we are dissecting.

Conclusion

In solving trigonometric equations, it is essential to recognize the periodic property of trigonometric functions. The use of general solutions efficiently expresses the infinite possibilities brought by these periodic functions. Equipped with this knowledge, dealing with any trigonometric equation becomes more systematic and predictable.


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General Solutions in Trigonometric Equations

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