Trigonometric Identities: Compound Angles, Sine, Cosine, and Tangent

Compound Angle and Angle Sum/Difference Rules

• Compound angle rules are essential for understanding trigonometric identities for sine, cosine, and tangent.
• These rules are important but can be found on the formula sheet, so memorization isn't necessary but familiarity is useful.

Example 1: Expanding Sine

• Problem: Expand ext{sine}(3m + 2k).
  • Rule Used: Sine addition formula: ext{sine}(a + b) = ext{sine}(a) ext{cos}(b) + ext{cos}(a) ext{sine}(b).
  • Solution:
  • Let a = 3m and b = 2k.
  • Expansion: ext{sine}(3m + 2k) = ext{sine}(3m) ext{cos}(2k) + ext{cos}(3m) ext{sine}(2k).
  • The formula cannot be simplified further at this level.

Example 2: Simplifying Cosine and Sine

• Problem: Simplify ext{cos}(27) ext{cos}(12) - ext{sine}(27) ext{sine}(12).
  • Observation: Repeated angles allow for applying the cosine subtraction formula: ext{cos}(a - b) = ext{cos}(a) ext{cos}(b) + ext{sine}(a) ext{sine}(b).
  • Solution:
  • This leads to: ext{cos}(27 + 12) = ext{cos}(39).

Example 3: Finding Exact Values of Cosine

• Problem: Find ext{cos}(15).
  • Approach: Use angles we know to express 15 in a useful way: ext{cos}(15) = ext{cos}(45 - 30).
  • Expansion: Using the cosine addition formula:
    ext{cos}(a - b) = ext{cos}(a) ext{cos}(b) + ext{sine}(a) ext{sine}(b)
    = ext{cos}(45) ext{cos}(30) + ext{sine}(45) ext{sine}(30) .
  • Exact Values:
  • ext{cos}(45) = rac{ ext{root } 2}{2}, ext{cos}(30) = rac{ ext{root } 3}{2}, ext{sine}(45) = rac{ ext{root } 2}{2}, ext{sine}(30) = rac{1}{2}.
  • Substitute values: rac{ ext{root } 2}{2} imes rac{ ext{root } 3}{2} + rac{ ext{root } 2}{2} imes rac{1}{2} = rac{ ext{root } 6}{4} + rac{ ext{root } 2}{4} = rac{ ext{root } 6 + ext{root } 2}{4} .

Example 4: Finding Exact Values of Sine in Radians

• Problem: Find ext{sine}igg(rac{ ext{pi}}{12}igg).
  • Approach: Represent as ext{sine}igg(rac{ ext{pi}}{3} - rac{ ext{pi}}{4}igg) and use sine subtraction formula.
  • Expansion:
    ext{sine}(a - b) = ext{sine}(a) ext{cos}(b) - ext{cos}(a) ext{sine}(b) .
    ext{sine}igg(rac{ ext{pi}}{3}igg) = rac{ ext{root } 3}{2}, ext{cos}igg(rac{ ext{pi}}{4}igg) = rac{ ext{root } 2}{2} etc.
  • Solution: Combine simplifications to yield the result similar to previous cosine calculations.

Example 5: Finding Tan Based on Sum of Angles

• Problem: Find ext{tan}igg(rac{7 ext{pi}}{12}igg).
  • Step: Rewrite as ext{tan}igg(rac{ ext{pi}}{4} + rac{ ext{pi}}{3}igg) .
  • Using Tan Addition Formula:
    ext{tan}(a + b) = rac{ ext{tan}(a) + ext{tan}(b)}{1 - ext{tan}(a) ext{tan}(b)} .
  • Exact Values: Fill in known values, rationalize the denominator for the final result.

Example 6: Using Trigonometric Identities

• Problem: Given ext{cos}(a) = rac{2}{3} and ext{sine}(b) = rac{24}{25}, find ext{sine}(a + b).
  • Formula Used:
    ext{sine}(a + b) = ext{sine}(a) ext{cos}(b) + ext{cos}(a) ext{sine}(b) .
  • Determine Remaining Values: Use Pythagorean identities to find ext{sine}(a) and ext{cos}(b).
  • Final Assembly: Combine to create a single fraction or separated fractions as preferred.

Example 7: Proof of the Identity

• Goal: Prove ext{cos}(x + rac{3 ext{pi}}{2}) = ext{sine}(x).
  • Process: Start from the left-hand side, use angle sum rules, simplify using known trigonometric values for specific angles.

Practice

• Complete exercises from sections 12l and 12m to reinforce these concepts.
• Questions may involve various applications of the rules outlined in these examples.