Worksheet A Topic 3.10 Part I Trigonometric Equations

Let's dive into trigonometric equations, a fundamental aspect of trigonometry with far-reaching applications in various fields. Understanding how to solve these equations is crucial for anyone studying mathematics, physics, engineering, or computer science. This article will thoroughly cover trigonometric equations, exploring their basic concepts, types, and solution methods.

Introduction to Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Solving these equations means finding the values of the variable (usually represented as x or θ) that make the equation true. Unlike trigonometric identities, which are true for all values of the variable, trigonometric equations are only true for specific values or a set of values.

Basic Concepts

Before diving into solving trigonometric equations, let’s review some fundamental concepts:

• Trigonometric Functions: Understand the definitions and properties of sin, cos, tan, cot, sec, and csc.
• Unit Circle: Familiarize yourself with the unit circle, which provides a visual representation of trigonometric function values for various angles.
• Periodicity: Recognize that trigonometric functions are periodic, meaning their values repeat at regular intervals. For example, sin(x) and cos(x) have a period of 2π, while tan(x) has a period of π.
• Inverse Trigonometric Functions: Know the definitions and ranges of inverse trigonometric functions like arcsin(x), arccos(x), and arctan(x).

Types of Trigonometric Equations

Trigonometric equations come in various forms, including:

• Simple Trigonometric Equations: Equations that involve only one trigonometric function, such as sin(x) = 0.5.
• Linear Trigonometric Equations: Equations that can be written in a linear form using trigonometric functions, such as 2sin(x) + 1 = 0.
• Quadratic Trigonometric Equations: Equations that involve trigonometric functions raised to the power of 2, such as sin²(x) - sin(x) = 0.
• Equations Involving Multiple Angles: Equations where the argument of the trigonometric function is a multiple of the variable, such as sin(2x) = 1.
• Equations Involving Sums or Differences of Angles: Equations that include trigonometric functions of sums or differences of angles, such as sin(x + π/4) = 0.
• Equations Involving Products of Trigonometric Functions: Equations that involve products of trigonometric functions, such as sin(x)cos(x) = 0.

Solving Simple Trigonometric Equations

Let's start with the simplest form of trigonometric equations.

General Approach

To solve simple trigonometric equations, follow these steps:

1. Isolate the Trigonometric Function: Rewrite the equation to isolate the trigonometric function on one side.
2. Find the Reference Angle: Determine the reference angle, which is the acute angle formed by the terminal side of the angle and the x-axis.
3. Identify Quadrants: Determine the quadrants in which the solutions lie based on the sign of the trigonometric function.
4. Find Solutions in the Interval [0, 2π): Find all solutions within one period (0 to 2π for sin, cos, csc, and sec, and 0 to π for tan and cot).
5. General Solutions: Write the general solutions by adding integer multiples of the period to the solutions found in the interval [0, 2π).

Examples

Example 1: Solve sin(x) = 0.5

1. Isolate the Trigonometric Function: The equation is already in the desired form: sin(x) = 0.5.

2. Find the Reference Angle: The reference angle α such that sin(α) = 0.5 is α = π/6 (30°).

3. Identify Quadrants: Since sin(x) is positive, the solutions lie in the first and second quadrants.

4. Find Solutions in the Interval [0, 2π):

   • In the first quadrant: x₁ = π/6
   • In the second quadrant: x₂ = π - π/6 = 5π/6

5. General Solutions: The general solutions are:

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

Example 2: Solve cos(x) = -√3/2

1. Isolate the Trigonometric Function: The equation is already in the desired form: cos(x) = -√3/2.

2. Find the Reference Angle: The reference angle α such that cos(α) = √3/2 is α = π/6 (30°).

3. Identify Quadrants: Since cos(x) is negative, the solutions lie in the second and third quadrants.

4. Find Solutions in the Interval [0, 2π):

   • In the second quadrant: x₁ = π - π/6 = 5π/6
   • In the third quadrant: x₂ = π + π/6 = 7π/6

5. General Solutions: The general solutions are:

   • x = 5π/6 + 2πk, where k is an integer
   • x = 7π/6 + 2πk, where k is an integer

Example 3: Solve tan(x) = 1

1. Isolate the Trigonometric Function: The equation is already in the desired form: tan(x) = 1.

2. Find the Reference Angle: The reference angle α such that tan(α) = 1 is α = π/4 (45°).

3. Identify Quadrants: Since tan(x) is positive, the solutions lie in the first and third quadrants.

4. Find Solutions in the Interval [0, 2π):

   • In the first quadrant: x₁ = π/4
   • In the third quadrant: x₂ = π + π/4 = 5π/4

5. General Solutions: Since the period of tan(x) is π, we can combine the solutions:

   • x = π/4 + πk, where k is an integer

Solving Linear Trigonometric Equations

Linear trigonometric equations are equations that can be expressed in a linear form involving trigonometric functions.

General Approach

To solve linear trigonometric equations, follow these steps:

1. Isolate the Trigonometric Function: Rewrite the equation to isolate the trigonometric function on one side.
2. Solve for the Trigonometric Function: Solve the equation for the trigonometric function.
3. Find the Reference Angle: Determine the reference angle.
4. Identify Quadrants: Determine the quadrants in which the solutions lie.
5. Find Solutions in the Interval [0, 2π): Find all solutions within one period.
6. General Solutions: Write the general solutions by adding integer multiples of the period.

Examples

Example 1: Solve 2sin(x) + 1 = 0

1. Isolate the Trigonometric Function:

   • 2sin(x) = -1
   • sin(x) = -1/2

2. Find the Reference Angle: The reference angle α such that sin(α) = 1/2 is α = π/6 (30°).

3. Identify Quadrants: Since sin(x) is negative, the solutions lie in the third and fourth quadrants.

4. Find Solutions in the Interval [0, 2π):

   • In the third quadrant: x₁ = π + π/6 = 7π/6
   • In the fourth quadrant: x₂ = 2π - π/6 = 11π/6

5. General Solutions: The general solutions are:

   • x = 7π/6 + 2πk, where k is an integer
   • x = 11π/6 + 2πk, where k is an integer

Example 2: Solve √3cos(x) - 1 = 0

1. Isolate the Trigonometric Function:

   • √3cos(x) = 1
   • cos(x) = 1/√3 = √3/3

2. Find the Reference Angle: The reference angle α such that cos(α) = √3/3 is α = arccos(√3/3) ≈ 0.955 radians (54.74°).

3. Identify Quadrants: Since cos(x) is positive, the solutions lie in the first and fourth quadrants.

4. Find Solutions in the Interval [0, 2π):

   • In the first quadrant: x₁ = arccos(√3/3) ≈ 0.955
   • In the fourth quadrant: x₂ = 2π - arccos(√3/3) ≈ 5.328

5. General Solutions: The general solutions are:

   • x = arccos(√3/3) + 2πk, where k is an integer
   • x = (2π - arccos(√3/3)) + 2πk, where k is an integer

Solving Quadratic Trigonometric Equations

Quadratic trigonometric equations are equations that involve trigonometric functions raised to the power of 2.

General Approach

To solve quadratic trigonometric equations, follow these steps:

1. Rewrite the Equation: Rewrite the equation in the form of a quadratic equation, such as asin²(x) + bsin(x) + c = 0.
2. Solve the Quadratic Equation: Solve the quadratic equation for the trigonometric function using factoring, completing the square, or the quadratic formula.
3. Solve for the Variable: Solve for the variable x using the methods for simple trigonometric equations.
4. Find Solutions in the Interval [0, 2π): Find all solutions within one period.
5. General Solutions: Write the general solutions by adding integer multiples of the period.

Examples

Example 1: Solve 2sin²(x) - sin(x) - 1 = 0

1. Rewrite the Equation: The equation is already in the quadratic form.

2. Solve the Quadratic Equation: Let y = sin(x). Then the equation becomes 2y² - y - 1 = 0.

   • Factoring: (2y + 1)(y - 1) = 0
   • So, y = -1/2 or y = 1
   • Thus, sin(x) = -1/2 or sin(x) = 1

3. Solve for the Variable:

   • For sin(x) = -1/2:

     • Reference angle: π/6
     • Quadrants: Third and Fourth
     • Solutions: x = 7π/6 + 2πk and x = 11π/6 + 2πk, where k is an integer

   • For sin(x) = 1:

     • Solution: x = π/2 + 2πk, where k is an integer

4. Find Solutions in the Interval [0, 2π):

   • x = 7π/6, 11π/6, π/2

5. General Solutions:

   • x = 7π/6 + 2πk, where k is an integer
   • x = 11π/6 + 2πk, where k is an integer
   • x = π/2 + 2πk, where k is an integer

Example 2: Solve cos²(x) - 3cos(x) + 2 = 0

1. Rewrite the Equation: The equation is already in the quadratic form.

2. Solve the Quadratic Equation: Let y = cos(x). Then the equation becomes y² - 3y + 2 = 0.

   • Factoring: (y - 1)(y - 2) = 0
   • So, y = 1 or y = 2
   • Thus, cos(x) = 1 or cos(x) = 2

3. Solve for the Variable:

   • For cos(x) = 1:

     • Solution: x = 0 + 2πk, where k is an integer

   • For cos(x) = 2:

     • No solution, since -1 ≤ cos(x) ≤ 1

4. Find Solutions in the Interval [0, 2π):

   • x = 0

5. General Solutions:

   • x = 2πk, where k is an integer

Equations Involving Multiple Angles

Equations involving multiple angles have the form of trigonometric functions with arguments that are multiples of the variable, such as sin(2x), cos(3x), etc.

General Approach

To solve equations involving multiple angles, follow these steps:

1. Solve for the Multiple Angle: Solve the equation for the trigonometric function of the multiple angle.
2. Find Solutions for the Multiple Angle: Find all solutions for the multiple angle within a suitable interval.
3. Solve for the Variable: Divide each solution by the coefficient of the variable to find the solutions for x.
4. Find Solutions in the Interval [0, 2π): Find all solutions within one period.
5. General Solutions: Write the general solutions by adding integer multiples of the period.

Examples

Example 1: Solve sin(2x) = 1/2

1. Solve for the Multiple Angle: The equation is already in the desired form: sin(2x) = 1/2.

2. Find Solutions for the Multiple Angle: Let θ = 2x. Then sin(θ) = 1/2.

   • Reference angle: π/6
   • Quadrants: First and Second
   • Solutions: θ = π/6 + 2πk and θ = 5π/6 + 2πk, where k is an integer

3. Solve for the Variable:

   • 2x = π/6 + 2πk => x = π/12 + πk
   • 2x = 5π/6 + 2πk => x = 5π/12 + πk, where k is an integer

4. Find Solutions in the Interval [0, 2π):

   • x = π/12, 13π/12, 5π/12, 17π/12

5. General Solutions:

   • x = π/12 + πk, where k is an integer
   • x = 5π/12 + πk, where k is an integer

Example 2: Solve cos(3x) = -√2/2

1. Solve for the Multiple Angle: The equation is already in the desired form: cos(3x) = -√2/2.

2. Find Solutions for the Multiple Angle: Let θ = 3x. Then cos(θ) = -√2/2.

   • Reference angle: π/4
   • Quadrants: Second and Third
   • Solutions: θ = 3π/4 + 2πk and θ = 5π/4 + 2πk, where k is an integer

3. Solve for the Variable:

   • 3x = 3π/4 + 2πk => x = π/4 + (2π/3)k
   • 3x = 5π/4 + 2πk => x = 5π/12 + (2π/3)k, where k is an integer

4. Find Solutions in the Interval [0, 2π):

   • x = π/4, 11π/12, 19π/12, 5π/12, 13π/12, 7π/4

5. General Solutions:

   • x = π/4 + (2π/3)k, where k is an integer
   • x = 5π/12 + (2π/3)k, where k is an integer

Strategies for Complex Trigonometric Equations

Some trigonometric equations require more advanced strategies, such as using trigonometric identities, substitution, or graphical methods.

Using Trigonometric Identities

Trigonometric identities are essential tools for simplifying and solving trigonometric equations. Common identities include:

• Pythagorean Identities: sin²(x) + cos²(x) = 1, 1 + tan²(x) = sec²(x), 1 + cot²(x) = csc²(x)
• Double Angle Identities: sin(2x) = 2sin(x)cos(x), cos(2x) = cos²(x) - sin²(x) = 2cos²(x) - 1 = 1 - 2sin²(x)
• Sum and Difference Identities: sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b), cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)

Example: Solve sin(2x) + cos(x) = 0

1. Use Trigonometric Identities: Apply the double angle identity sin(2x) = 2sin(x)cos(x).

   • 2sin(x)cos(x) + cos(x) = 0

2. Factor:

   • cos(x)(2sin(x) + 1) = 0

3. Solve for Each Factor:

   • cos(x) = 0 or 2sin(x) + 1 = 0

4. Solve for the Variable:

   • For cos(x) = 0:

     • x = π/2 + πk, where k is an integer

   • For 2sin(x) + 1 = 0 => sin(x) = -1/2:

     • x = 7π/6 + 2πk and x = 11π/6 + 2πk, where k is an integer

5. Find Solutions in the Interval [0, 2π):

   • x = π/2, 3π/2, 7π/6, 11π/6

6. General Solutions:

   • x = π/2 + πk, where k is an integer
   • x = 7π/6 + 2πk, where k is an integer
   • x = 11π/6 + 2πk, where k is an integer

Substitution Techniques

Substitution can simplify complex trigonometric equations by replacing trigonometric functions with variables.

Example: Solve tan²(x) + 3tan(x) + 2 = 0

1. Substitution: Let y = tan(x).

   • y² + 3y + 2 = 0

2. Solve the Quadratic Equation:

   • (y + 1)(y + 2) = 0
   • y = -1 or y = -2

3. Substitute Back:

   • tan(x) = -1 or tan(x) = -2

4. Solve for the Variable:

   • For tan(x) = -1:

     • x = 3π/4 + πk, where k is an integer

   • For tan(x) = -2:

     • x = arctan(-2) + πk ≈ -1.107 + πk, where k is an integer

5. Find Solutions in the Interval [0, 2π):

   • x = 3π/4, 7π/4, 2.034, 5.176

6. General Solutions:

   • x = 3π/4 + πk, where k is an integer
   • x = arctan(-2) + πk, where k is an integer

Graphical Methods

Graphical methods can be used to approximate solutions to trigonometric equations, especially when analytical solutions are difficult to find.

1. Rewrite the Equation: Rewrite the equation in the form f(x) = g(x).
2. Graph the Functions: Graph y = f(x) and y = g(x) on the same coordinate plane.
3. Find Intersection Points: Identify the x-coordinates of the intersection points, which are the approximate solutions to the equation.

Conclusion

Mastering trigonometric equations is crucial for understanding and applying trigonometry in various fields. By understanding the basic concepts, types of equations, and solution methods, you can confidently tackle a wide range of problems. Remember to practice regularly and utilize trigonometric identities and other strategies to simplify complex equations.