Mastering Sum and Difference Formulas in Trigonometry

PreAlgebra Grades High School 6:57 Video
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Lesson Description

This lesson delves into the sum and difference formulas for sine, cosine, and tangent, enabling students to calculate exact trigonometric values for angles not readily available on the unit circle. Students will learn to apply these formulas, condense trigonometric expressions, and solve problems involving angles in specific quadrants.

Video Resource

Sum and Difference Formulas

Mario's Math Tutoring

Duration: 6:57
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Key Concepts

  • Sum and difference formulas for sine, cosine, and tangent.
  • Unit circle values for common angles (multiples of 30, 45, and 60 degrees).
  • Radian and degree conversions.
  • Trigonometric functions in different quadrants.
  • Simplifying trigonometric expressions.

Learning Objectives

  • Students will be able to apply the sum and difference formulas to calculate the exact values of trigonometric functions for non-standard angles.
  • Students will be able to condense trigonometric expressions using the sum and difference formulas.
  • Students will be able to determine the sign of trigonometric functions based on the quadrant of the angle.
  • Students will be able to convert between radians and degrees.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the unit circle and the values of sine, cosine, and tangent for common angles (0, π/6, π/4, π/3, π/2, etc.). Briefly introduce the concept of sum and difference formulas as tools for finding exact values of trigonometric functions for angles not directly on the unit circle.
  • Sine Sum and Difference Formulas (15 mins)
    Introduce the sine sum and difference formulas: sin(A + B) = sinAcosB + cosAsinB and sin(A - B) = sinAcosB - cosAsinB. Work through Example 1 from the video (finding sin(7π/12)) step-by-step, emphasizing the conversion to degrees, identifying suitable angles from the unit circle (60 and 45 degrees), and substituting into the formula. Stress the importance of knowing exact values from the unit circle.
  • Cosine Sum and Difference Formulas (15 mins)
    Introduce the cosine sum and difference formulas: cos(A + B) = cosAcosB - sinAsinB and cos(A - B) = cosAcosB + sinAsinB. Work through Example 2 from the video (condensing a cosine sum expression), demonstrating how to recognize the right-hand side of the formula and simplify it to the left-hand side. Emphasize the sign change in the cosine formulas.
  • Tangent Sum and Difference Formulas (20 mins)
    Introduce the tangent sum and difference formulas: tan(A + B) = (tanA + tanB) / (1 - tanAtanB) and tan(A - B) = (tanA - tanB) / (1 + tanAtanB). Work through Example 3 from the video (finding tan(U - V) given sin(U) and cos(V) and the quadrants of U and V). Stress the importance of drawing the triangles in the correct quadrants, determining the signs of the sides, and using the Pythagorean theorem to find the missing side. Guide students through the arithmetic simplification of the complex fraction.
  • Practice Problems (15 mins)
    Provide students with practice problems covering all three formulas. Encourage them to work in pairs or small groups. Circulate to provide assistance and answer questions.

Interactive Exercises

  • Angle Matching
    Provide a list of angles (e.g., 15°, 75°, 105°, 165°) and have students match them to expressions using sum and difference formulas (e.g., 45° - 30°, 45° + 30°, 60° + 45°, 180° - 15°).
  • Formula Scramble
    Write the sum and difference formulas on strips of paper, cutting each formula into individual terms. Have students reassemble the formulas correctly.

Discussion Questions

  • Why are the sum and difference formulas useful?
  • How does the quadrant of an angle affect the sign of its trigonometric functions?
  • How do you decide which two angles to use when finding the trigonometric value of an angle using the sum or difference formulas?

Skills Developed

  • Application of trigonometric identities.
  • Problem-solving skills in trigonometry.
  • Analytical thinking.
  • Understanding of the unit circle and trigonometric relationships.
  • Algebraic manipulation.

Multiple Choice Questions

Question 1:

Which of the following is the correct formula for sin(A + B)?

Correct Answer: sinAcosB + cosAsinB

Question 2:

What is the value of cos(π/12) using sum/difference formulas, given π/12 = π/3 - π/4?

Correct Answer: (√6 + √2)/4

Question 3:

Which quadrant is an angle θ in if sin(θ) < 0 and cos(θ) < 0?

Correct Answer: Quadrant III

Question 4:

What is the simplified form of cos(A)cos(B) + sin(A)sin(B)?

Correct Answer: cos(A - B)

Question 5:

Which of the following is the formula for tan(A - B)?

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

Question 6:

If sin(U) = 3/5 and U is in Quadrant II, what is cos(U)?

Correct Answer: -4/5

Question 7:

What is the value of sin(75°) given 75° = 45° + 30°?

Correct Answer: (√6 + √2)/4

Question 8:

Which of the following is equivalent to sin(π/2 + x)?

Correct Answer: -sin(x)

Question 9:

Given tan(A) = 1 and tan(B) = 2, what is tan(A+B)?

Correct Answer: -3

Question 10:

If cos(V) = 5/13 and V is in Quadrant IV, what is sin(V)?

Correct Answer: -12/13

Fill in the Blank Questions

Question 1:

The formula for cos(A - B) is cos(A)cos(B) ___ sin(A)sin(B).

Correct Answer: +

Question 2:

To use the sum and difference formulas effectively, you must know the trigonometric values of angles from the ____ circle.

Correct Answer: unit

Question 3:

The angle 7π/12 can be expressed as the sum of π/3 and ____.

Correct Answer: π/4

Question 4:

The tangent function is opposite over ____.

Correct Answer: adjacent

Question 5:

If an angle is in Quadrant III, both its sine and cosine are ____.

Correct Answer: negative

Question 6:

The formula for sin(A + B) is sin(A)cos(B) + cos(A)sin(___).

Correct Answer: B

Question 7:

When using sum and difference formulas, always look to express angles in terms of multiples of 30, 45, and ___ degrees.

Correct Answer: 60

Question 8:

Converting radians to degrees, π radians is equal to ___ degrees.

Correct Answer: 180

Question 9:

In the sum and difference formulas, the cosine sum formula cos(A + B) is equal to cosAcosB ___ sinAsinB.

Correct Answer: -

Question 10:

In a right triangle, the hypotenuse is always considered ____.

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.T.2.2:[Objective] Apply the Law of Sines and Law of Cosines to solve problems.

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