Mastering Double Angle Formulas: A PreCalculus Exploration

PreAlgebra Grades High School 10:04 Video
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Lesson Description

Explore and apply double angle formulas for sine, cosine, and tangent, including solving trigonometric equations. This lesson uses real-world examples and practice problems to solidify understanding.

Video Resource

Double Angle Formulas - How to Use (2 Examples)

Mario's Math Tutoring

Duration: 10:04
Watch on YouTube

Key Concepts

  • Double Angle Formulas (Sine, Cosine, Tangent)
  • Trigonometric Identities
  • Solving Trigonometric Equations
  • Unit Circle

Learning Objectives

  • Students will be able to apply double angle formulas to calculate trigonometric values.
  • Students will be able to solve trigonometric equations using double angle formulas and factoring techniques.
  • Students will be able to verify solutions using trigonometric identities.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the basic trigonometric functions (sine, cosine, tangent) and their relationships. Introduce the concept of double angle formulas and their importance in simplifying trigonometric expressions and solving equations. Show the video to introduce the concept.
  • Example 1: Calculating Trigonometric Values (15 mins)
    Work through the first example from the video, focusing on how to determine the values of sin(2θ), cos(2θ), and tan(2θ) given the value of sin(θ) and the quadrant in which θ lies. Emphasize the use of the Pythagorean theorem to find missing side lengths and the importance of considering the sign of trigonometric functions in different quadrants.
  • Example 2: Solving Trigonometric Equations (20 mins)
    Work through the second example from the video, demonstrating how to solve a trigonometric equation involving cos(2x). Highlight the different options for the double angle formula of cosine and explain the strategic choice of the formula that allows for factoring. Guide students through the factoring process and the application of the zero product property to find the solutions. Discuss the concept of general solutions and how to express them using multiples of 2π/3.
  • Practice Problems (15 mins)
    Provide students with practice problems that require them to apply the double angle formulas and solve trigonometric equations. Encourage them to work individually or in small groups and provide assistance as needed.
  • Wrap-up and Review (5 mins)
    Summarize the key concepts covered in the lesson, including the double angle formulas and their applications. Answer any remaining questions and provide a brief overview of what will be covered in the next lesson.

Interactive Exercises

  • Double Angle Formula Matching
    Provide students with a list of double angle formulas and a list of expanded expressions. Have them match the formulas with their corresponding expressions.
  • Trigonometric Equation Scavenger Hunt
    Hide trigonometric equations around the classroom and have students find them and solve them using double angle formulas.

Discussion Questions

  • Why are there multiple versions of the double angle formula for cosine, and how do you decide which one to use?
  • How does the quadrant in which the angle lies affect the sign of the trigonometric functions?

Skills Developed

  • Application of Trigonometric Identities
  • Problem-Solving
  • Algebraic Manipulation
  • Critical Thinking

Multiple Choice Questions

Question 1:

Which of the following is the correct double angle formula for sin(2θ)?

Correct Answer: 2sin(θ)cos(θ)

Question 2:

Which of the following is a double angle formula for cos(2θ)?

Correct Answer: cos²(θ) - sin²(θ)

Question 3:

What is the double angle formula for tan(2θ)?

Correct Answer: (2tan(θ))/(1 - tan²(θ))

Question 4:

If sin(θ) = 3/5 and θ is in the first quadrant, what is sin(2θ)?

Correct Answer: 24/25

Question 5:

If cos(θ) = -4/5 and θ is in the second quadrant, what is cos(2θ)?

Correct Answer: -7/25

Question 6:

Given tan(θ) = 1, what is tan(2θ)?

Correct Answer: Undefined

Question 7:

Which identity is most useful when solving the equation cos(2x) - cos(x) = 0?

Correct Answer: cos(2x) = 2cos²(x) - 1

Question 8:

If θ is in the second quadrant and sin(θ) = 12/13, find the value of cos(2θ).

Correct Answer: 25/169

Question 9:

Simplify the expression: 2sin(x/2)cos(x/2)

Correct Answer: sin(x)

Question 10:

Which equation is equivalent to cos(2θ) + 1?

Correct Answer: 2cos²(θ)

Fill in the Blank Questions

Question 1:

The double angle formula for sine is sin(2θ) = 2sin(θ)________(θ).

Correct Answer: cos

Question 2:

cos(2θ) = ________(θ) - sin²(θ).

Correct Answer: cos²

Question 3:

tan(2θ) = (2tan(θ)) / (1 - tan²(________)).

Correct Answer: θ

Question 4:

If cos(θ) = 1/2, then cos(2θ) = 2(1/2)² - 1 = ________.

Correct Answer: -1/2

Question 5:

To solve cos(2x) - cos(x) = 0, one strategy is to replace cos(2x) with 2cos²(x) - ________.

Correct Answer: 1

Question 6:

If sin(θ) = 0.6, then 1 - 2sin²(θ) = 1 - 2(0.6)² = ________.

Correct Answer: 0.28

Question 7:

Given tan(θ) = 3, tan(2θ) = (2 * 3)/(1-3²) = ________.

Correct Answer: -3/4

Question 8:

Another form of the cosine double angle formula is cos(2x)= 1-2________(x).

Correct Answer: sin²

Question 9:

sin(2x) is also known as 2sin(x)________(x).

Correct Answer: cos

Question 10:

The general solution accounts for all solutions, usually in increments of ____π.

Correct Answer: 2

Educational Standards

PC.T.2.1:[Objective] Create models for situations involving trigonometry. PC.F.3.3:[Objective] Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.T.2:[Standard] Apply trigonometric concepts beyond the right triangle.

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