Mastering Trigonometric Equations: General Solutions and Interval [0, 2π)

PreAlgebra Grades High School 19:05 Video
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Lesson Description

Learn to solve trigonometric equations, find general solutions, and identify solutions within the interval [0, 2π) using the unit circle and factoring techniques.

Video Resource

Solving Trig Equations (General Solution and [0,2pi))

Mario's Math Tutoring

Duration: 19:05
Watch on YouTube

Key Concepts

  • Unit Circle
  • General Solutions of Trigonometric Equations
  • Solutions in the Interval [0, 2π)
  • Trigonometric Identities
  • Factoring Trigonometric Equations
  • Multiple Angle Equations

Learning Objectives

  • Solve trigonometric equations using the unit circle.
  • Express general solutions for trigonometric equations.
  • Identify solutions of trigonometric equations within the interval [0, 2π).
  • Solve trigonometric equations involving multiple angles.
  • Solve trigonometric equations by factoring.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the unit circle and its significance in solving trigonometric equations. Explain the difference between general solutions and solutions within a specific interval [0, 2π).
  • Example 1: Solving sin(x) = 1/2 (7 mins)
    Demonstrate how to find the solutions to sin(x) = 1/2 within the interval [0, 2π) using the unit circle. Explain how to express the general solution using the periodicity of the sine function.
  • Example 2: Solving cos(x) = -√3/2 (7 mins)
    Demonstrate how to find the solutions to cos(x) = -√3/2 within the interval [0, 2π) using the unit circle. Explain how to express the general solution using the periodicity of the cosine function.
  • Example 3: Solving 2sin(x) + √3 = 0 (8 mins)
    Walk through the steps to isolate sin(x) and then find the solutions within the interval [0, 2π). Explain how to express the general solution.
  • Example 4: Solving tan(x) = 1 (8 mins)
    Demonstrate how to find the solutions to tan(x) = 1 within the interval [0, 2π) using the unit circle. Explain how to express the general solution using the periodicity of the tangent function. Emphasize the different period of tan(x).
  • Example 5: Solving sin(2x) = √3/2 (10 mins)
    Solve a trigonometric equation involving a multiple angle (2x). Find the solutions for 2x and then divide by 2 to find the solutions for x within the interval [0, 2π). Explain how the general solution is affected by the multiple angle.
  • Example 6: Solving 2cos²(x) - cos(x) - 1 = 0 (10 mins)
    Solve a trigonometric equation by factoring. Substitute 'u' for cos(x) to make it easier to factor. Find the solutions for cos(x) and then find the corresponding angles within the interval [0, 2π). Explain how to express the general solution.
  • Conclusion (5 mins)
    Summarize the key concepts covered in the video: solving trigonometric equations using the unit circle, finding general solutions, and finding solutions within the interval [0, 2π). Reinforce the importance of understanding trigonometric identities and factoring techniques.

Interactive Exercises

  • Unit Circle Practice
    Students label the unit circle with degree and radian measures for special angles and find the sine, cosine, and tangent values for each angle.
  • Solving Equations Worksheet
    Students work individually or in groups to solve a variety of trigonometric equations, including those requiring factoring and those with multiple angles. Check answers and discuss any difficulties.

Discussion Questions

  • How does the unit circle help in visualizing and solving trigonometric equations?
  • What is the significance of the general solution of a trigonometric equation?
  • Explain the effect of a multiple angle (e.g., 2x, 3x) on the solutions of a trigonometric equation.
  • When solving trigonometric equations, how do we use trigonometric identities to simplify the equation?
  • Discuss the similarities and differences between solving algebraic equations and trigonometric equations.

Skills Developed

  • Problem-solving
  • Analytical Thinking
  • Application of Trigonometric Identities
  • Understanding of the Unit Circle

Multiple Choice Questions

Question 1:

What is the general solution for sin(x) = 0?

Correct Answer: x = nπ, where n is an integer

Question 2:

What are the solutions for cos(x) = 1/2 in the interval [0, 2π)?

Correct Answer: π/3, 5π/3

Question 3:

What is the period of the tangent function?

Correct Answer: π

Question 4:

If sin(2x) = 1, what is the value of x in the interval [0, π)?

Correct Answer: π/4

Question 5:

Which of the following is a solution to 2cos²(x) - 1 = 0 in the interval [0, 2π)?

Correct Answer: π/4

Question 6:

The general solution of cos(x) = 0 is:

Correct Answer: x = (π/2) + nπ

Question 7:

What are the solutions to tan(x) = -1 in the interval [0, 2π)?

Correct Answer: 3π/4, 7π/4

Question 8:

What is the general solution for sin(x) = 1?

Correct Answer: x = π/2 + 2nπ

Question 9:

For what values of x does cos(2x) = 0 in the interval [0, π)?

Correct Answer: π/4, 3π/4

Question 10:

Which of the following trigonometric identities is useful for solving 2sin²(x) + sin(x) - 1 = 0?

Correct Answer: Factoring

Fill in the Blank Questions

Question 1:

The general solution to sin(x) = -1 is x = __________ + 2nπ, where n is an integer.

Correct Answer: 3π/2

Question 2:

The solutions to cos(x) = 0 in the interval [0, 2π) are x = __________ and x = __________.

Correct Answer: π/2, 3π/2

Question 3:

The period of the sine and cosine functions is __________.

Correct Answer: 2π

Question 4:

If tan(x) = 0, then x = __________ + nπ, where n is an integer.

Correct Answer: nπ

Question 5:

To solve 2cos²(x) - cos(x) = 0, you can __________ cos(x) from the expression.

Correct Answer: factor

Question 6:

The interval [0, 2π) represents one complete revolution on the __________.

Correct Answer: unit circle

Question 7:

When solving sin(2x) = 1, you must divide your solutions for 2x by __________ to find the solutions for x.

Correct Answer: 2

Question 8:

The reference angle for 7π/6 is __________.

Correct Answer: π/6

Question 9:

The range of the cosine function is __________.

Correct Answer: [-1, 1]

Question 10:

When finding general solutions, 'n' represents any __________.

Correct Answer: integer

Educational Standards

PC.T.1:Make sense of the unit circle and its relationship to the graphs of trigonometric functions. PC.T.1.4:Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for 𝜋 − 𝑥, 𝜋 + 𝑥, and 2𝜋 − 𝑥 in terms of their values for x, where x is any real number. PC.T.2.1:[Objective] Create models for situations involving trigonometry.

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