Unlock Exact Trig Values: Mastering Sum and Difference Formulas

Algebra 2 Grades High School 19:25 Video
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Lesson Description

Learn to calculate exact values of trigonometric functions using sum and difference formulas. This lesson uses the unit circle to tackle angles beyond the standard values.

Video Resource

Sum and Difference Formulas for Sine, Cosine, and Tangent to Find Exact Values

Mario's Math Tutoring

Duration: 19:25
Watch on YouTube

Key Concepts

  • Sum and difference formulas for sine, cosine, and tangent
  • Unit circle values for common angles
  • Radian and degree conversions
  • Simplifying radical expressions

Learning Objectives

  • Apply sum and difference formulas to find exact values of trigonometric functions.
  • Convert between radians and degrees.
  • Simplify trigonometric expressions involving radicals.
  • Utilize the unit circle to determine trigonometric values of standard angles.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the unit circle and its key angles (30, 45, 60, 90 degrees and their radian equivalents). Briefly discuss the need for sum and difference formulas when dealing with angles not directly on the unit circle.
  • Video Presentation (15 mins)
    Play the Mario's Math Tutoring video. Instruct students to take notes on the sum and difference formulas for sine, cosine, and tangent. Emphasize the importance of correctly identifying 'a' and 'b' in the formulas.
  • Guided Practice (20 mins)
    Work through the first three examples from the video with the class, pausing at each step to explain the reasoning. Encourage students to participate by asking questions and suggesting steps. Focus on converting to degrees if students struggle with radians initially.
  • Independent Practice (20 mins)
    Have students work in pairs or individually on the remaining three examples from the video. Circulate to provide assistance and answer questions. After 10 minutes, have them switch the problems with their partner and have them solve the inverse problem. (i.e. if they did sine, have them do cosine.)
  • Wrap-up and Assessment (10 mins)
    Review the solutions to the last three examples. Administer the multiple-choice or fill-in-the-blank quiz to assess student understanding.

Interactive Exercises

  • Angle Matching
    Provide a list of angles (e.g., 105°, 15°, 195°) and have students match them with possible sum or difference combinations (e.g., 60° + 45°, 45° - 30°, 150° + 45°).
  • Formula Application
    Give students specific trigonometric expressions (e.g., sin(A+B), cos(A-B)) and have them expand the expressions using the appropriate sum or difference formula.

Discussion Questions

  • Why are sum and difference formulas useful?
  • How do you decide which two angles to use when applying the formulas?
  • What are common mistakes to avoid when using these formulas?
  • How does knowing the unit circle help with these problems?

Skills Developed

  • Application of trigonometric identities
  • Problem-solving with the unit circle
  • Algebraic manipulation and simplification

Multiple Choice Questions

Question 1:

The sum formula for sin(A + B) is:

Correct Answer: sin(A)cos(B) + cos(A)sin(B)

Question 2:

Which of the following is equal to cos(105°)?

Correct Answer: cos(60° + 45°)

Question 3:

The tangent difference formula is given by?

Correct Answer: (tanA - tanB)/(1 + tanAtanB)

Question 4:

To find the exact value of sin(15°), which combination is most appropriate?

Correct Answer: sin(45° - 30°)

Question 5:

What is the reference angle we use to solve for tan(285°)?

Correct Answer: 30°

Question 6:

What is the exact value of cos(π/4)?

Correct Answer: √2/2

Question 7:

In the formula for sin(A - B), if A = 90° and B = 30°, what is sin(A)?

Correct Answer: 1

Question 8:

Which formula is used to expand the value of Cos(A + B)?

Correct Answer: Cos(A)Cos(B) - Sin(A)Sin(B)

Question 9:

What is the exact value of Sin(30°)?

Correct Answer: 1/2

Question 10:

What is Sin(45°) - Cos(45°)?

Correct Answer: 0

Fill in the Blank Questions

Question 1:

The sum formula for cosine, cos(A + B), is cos(A)cos(B) ______ sin(A)sin(B).

Correct Answer: - (minus)

Question 2:

To find the exact value of tan(75°), you could use tan(45° + ______°).

Correct Answer: 30

Question 3:

The y-coordinate on the unit circle represents the ______ value of an angle.

Correct Answer: sine

Question 4:

Converting radians to degrees involves multiplying by ______ / π.

Correct Answer: 180

Question 5:

The tangent of an angle is found by dividing ______ over cosine.

Correct Answer: sine

Question 6:

The formula for sin(A - B) involves _______ sign between the terms.

Correct Answer: subtraction

Question 7:

To simplify radical expressions, you may need to ______ the denominator.

Correct Answer: rationalize

Question 8:

An angle of 120° is equivalent to ______π / 3 radians.

Correct Answer: 2

Question 9:

cos(60°) = _______

Correct Answer: 1/2

Question 10:

Exact trigonometric values are written without using ________

Correct Answer: decimals

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together). A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

Teaching Materials

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