Unlocking Half-Angle Formulas: Mastering Trigonometric Identities

PreAlgebra Grades High School 7:07 Video
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Lesson Description

Explore half-angle formulas for sine, cosine, and tangent, and learn how to apply them to solve trigonometric problems. This lesson covers simplifying expressions and finding exact values.

Video Resource

Half Angle Formulas - How to Use

Mario's Math Tutoring

Duration: 7:07
Watch on YouTube

Key Concepts

  • Half-Angle Formulas for Sine, Cosine, and Tangent
  • Unit Circle Values
  • Simplifying Trigonometric Expressions
  • Exact vs. Approximate Values

Learning Objectives

  • State and apply the half-angle formulas for sine, cosine, and tangent.
  • Simplify trigonometric expressions using half-angle formulas.
  • Determine the correct sign (positive or negative) when using half-angle formulas.
  • Calculate exact trigonometric values for angles that are half of common unit circle angles.

Educator Instructions

  • Introduction to Half-Angle Formulas (5 mins)
    Briefly review the unit circle and trigonometric identities. Introduce the half-angle formulas for sine, cosine, and tangent. Emphasize the importance of understanding when to use the positive or negative sign.
  • Example 1: Secant of Half an Angle (7 mins)
    Work through the first example in the video, finding the secant of theta/2. Highlight the steps of using Pythagorean theorem, converting secant to cosine, and simplifying the expression. Explain how to determine the correct sign.
  • Example 2: Cotangent of Half an Angle (5 mins)
    Demonstrate how to find the cotangent of theta/2 using the half-angle formula. Show how to manipulate the formula and simplify the expression.
  • Example 3: Simplifying with Cosine Half-Angle Identity (5 mins)
    Illustrate how to simplify a given expression using the cosine half-angle formula.
  • Example 4: Sine of 112.5 Degrees (7 mins)
    Calculate the sine of 112.5 degrees using the half-angle formula. Focus on determining the correct sign based on the quadrant of the angle.
  • Example 5: Cosine of 3pi/8 (6 mins)
    Calculate the cosine of 3pi/8 using the cosine half-angle formula. Emphasize the simplification process.
  • Practice Problems and Review (10 mins)
    Provide students with practice problems to work on individually or in small groups. Review the key concepts and address any remaining questions.

Interactive Exercises

  • Half-Angle Formula Matching
    Match trigonometric expressions with their simplified forms using half-angle formulas.
  • Sign Determination Practice
    Determine the correct sign (positive or negative) for various half-angle expressions given the quadrant of the angle.

Discussion Questions

  • When would you choose to use a half-angle formula instead of a sum or difference formula?
  • How does the quadrant of the angle affect the sign of the trigonometric function when using half-angle formulas?
  • Why is it important to know the difference between exact and approximate values when solving trigonometric expressions?

Skills Developed

  • Applying Trigonometric Identities
  • Simplifying Algebraic Expressions
  • Problem-Solving
  • Analytical Thinking

Multiple Choice Questions

Question 1:

The half-angle formula for sin(θ/2) is:

Correct Answer: ±√(1 - cos θ)/2

Question 2:

The half-angle formula for cos(θ/2) is:

Correct Answer: ±√(1 + cos θ)/2

Question 3:

Which quadrant is 112.5° located in?

Correct Answer: Quadrant II

Question 4:

To find the exact value of cos(π/8) using half-angle formulas, you would use half of what angle?

Correct Answer: π/4

Question 5:

When finding sin(θ/2), the sign of the result depends on the quadrant of:

Correct Answer: θ/2

Question 6:

What is the simplified form of √((1 + cos(4x))/2)?

Correct Answer: cos(2x)

Question 7:

The half-angle formula for tan(θ/2) is:

Correct Answer: sin(θ)/(1 - cos(θ))

Question 8:

If θ is in the third quadrant, what is the sign of sin(θ/2)?

Correct Answer: Positive

Question 9:

If given sec(θ/2), what would your first step likely be to use the half angle formula?

Correct Answer: Find the reciprocal

Question 10:

The expression √((2 - √2)/4) is equal to:

Correct Answer: cos(π/8)

Fill in the Blank Questions

Question 1:

The half-angle formula for sine is ±√(1 - cos θ)/____.

Correct Answer: 2

Question 2:

The cosine of 225° is -√____/2.

Correct Answer: 2

Question 3:

The half-angle formula for tangent, tan(θ/2) = sin(θ) / (1 - ____(θ)).

Correct Answer: cos

Question 4:

Before applying the formula you must determine if the sign is positive or ____.

Correct Answer: negative

Question 5:

The angle 3π/8 is half of the angle ____/4.

Correct Answer: 3pi

Question 6:

Cotangent is the ____ of tangent.

Correct Answer: reciprocal

Question 7:

Secant is the reciprocal of ____.

Correct Answer: cosine

Question 8:

Exact values from the unit circle are not ____.

Correct Answer: approximations

Question 9:

There are ____ minutes in one degree.

Correct Answer: 60

Question 10:

When theta is in the second quadrant, sine is ____.

Correct Answer: positive

Educational Standards

PC.T.2.1:[Objective] Create models for situations involving trigonometry. PC.T.1.4:[Objective] 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.F.3.3:[Objective] Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

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