Mastering Tangent Sum and Difference Formulas

Algebra 2 Grades High School 8:43 Video
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Lesson Description

This lesson will guide you through understanding and applying tangent sum and difference formulas, using examples in both degrees and radians. We'll reinforce your knowledge of the unit circle and unit conversions.

Video Resource

Tangent Sum and Difference Formulas How to Use

Mario's Math Tutoring

Duration: 8:43
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Key Concepts

  • Tangent Sum Formula: tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
  • Tangent Difference Formula: tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
  • Unit Circle and Trigonometric Values
  • Radian and Degree Conversions

Learning Objectives

  • Students will be able to state and apply the tangent sum and difference formulas.
  • Students will be able to evaluate trigonometric functions using the unit circle.
  • Students will be able to convert between degrees and radians.
  • Students will be able to simplify expressions involving tangent sum and difference formulas.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the unit circle and the values of tangent at key angles (0, π/6, π/4, π/3, π/2, etc.). Briefly discuss the need for sum and difference formulas when dealing with angles that are not directly on the unit circle.
  • Tangent Sum and Difference Formulas (10 mins)
    Introduce the tangent sum and difference formulas. Explain the difference between the two and emphasize the importance of using the correct formula based on whether it's a sum or difference of angles. Write the formulas on the board: tan(A + B) = (tan A + tan B) / (1 - tan A tan B) and tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
  • Example 1: Degrees (15 mins)
    Work through an example using degrees. For instance, find the value of tan(75°). Show how to break down 75° into 45° + 30°. Evaluate tan(45°) and tan(30°) using the unit circle. Substitute these values into the tangent sum formula and simplify. Clearly show each step of the simplification process.
  • Example 2: Radians (15 mins)
    Work through an example using radians. For instance, find the value of tan(π/12). Show how to express π/12 as a difference of two common radian angles, such as π/3 - π/4. Evaluate tan(π/3) and tan(π/4) using the unit circle. Substitute these values into the tangent difference formula and simplify. Again, demonstrate each step of the simplification process carefully.
  • Practice Problems (10 mins)
    Provide students with practice problems involving both degrees and radians. Encourage them to work independently or in pairs. Circulate to provide assistance as needed.
  • Wrap-up (5 mins)
    Review the key concepts and formulas. Answer any remaining questions. Preview the upcoming topics.

Interactive Exercises

  • Unit Circle Review
    Have students complete a blank unit circle, filling in the angles in radians and degrees, as well as the sine, cosine, and tangent values for key angles.
  • Formula Application Practice
    Provide students with a worksheet containing various angles (in both degrees and radians) that can be expressed as sums or differences of common angles. Have them use the tangent sum and difference formulas to evaluate these angles.

Discussion Questions

  • Why are the sum and difference formulas necessary when we have the unit circle?
  • How does the sign in the tangent sum formula differ from the tangent difference formula, and why is this important?
  • Can you think of other angles (besides the examples in the video) that could be easily evaluated using these formulas?

Skills Developed

  • Application of Trigonometric Identities
  • Unit Circle Mastery
  • Algebraic Manipulation
  • Problem-Solving

Multiple Choice Questions

Question 1:

What is the tangent sum formula for tan(A + B)?

Correct Answer: (tan A + tan B) / (1 - tan A tan B)

Question 2:

What is the tangent difference formula for tan(A - B)?

Correct Answer: (tan A - tan B) / (1 + tan A tan B)

Question 3:

What is the value of tan(π/4)?

Correct Answer: 1

Question 4:

What is the value of tan(π/6)?

Correct Answer: √3/3

Question 5:

Which of the following is a possible way to express 15° using common angles?

Correct Answer: 45° - 30°

Question 6:

Which of the following is a possible way to express π/12 using common angles?

Correct Answer: π/4 - π/6

Question 7:

To find tan(105°), you could use the formula with A = 60° and B = ?

Correct Answer: 45°

Question 8:

What is the value of tan(π/3)?

Correct Answer: √3

Question 9:

When using the tangent sum and difference formulas, it's important to know tangent values from the...

Correct Answer: Unit Circle

Question 10:

What is the first step when simplifying the result of the tangent sum/difference formulas?

Correct Answer: Substituting values

Fill in the Blank Questions

Question 1:

The tangent _____ formula is tan(A + B) = (tan A + tan B) / (1 - tan A tan B).

Correct Answer: sum

Question 2:

The tangent _____ formula is tan(A - B) = (tan A - tan B) / (1 + tan A tan B).

Correct Answer: difference

Question 3:

tan(45°) is equal to _____.

Correct Answer: 1

Question 4:

tan(30°) is equal to _____.

Correct Answer: √3/3

Question 5:

To evaluate tan(75°), you can rewrite it as tan(45° + _____).

Correct Answer: 30°

Question 6:

To evaluate tan(π/12), you can rewrite it as tan(π/3 - _____).

Correct Answer: π/4

Question 7:

tan(π/3) is equal to _____.

Correct Answer: √3

Question 8:

The unit circle is useful because it contains values of trigonometric functions for common ____.

Correct Answer: angles

Question 9:

When using the tangent sum formula, if tan A tan B = 1, the denominator becomes _____.

Correct Answer: 0

Question 10:

To simplify the final fraction from the tangent sum/difference formula, you might have to multiply by the _____.

Correct Answer: conjugate

Educational Standards

A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

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