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:
  • $ext{cos}(45) = \frac{ ext{root } 2}{2}$, $ext{cos}(30) = \frac{ ext{root } 3}{2}$, $ext{sine}(45) = \frac{ ext{root } 2}{2}$, $ext{sine}(30) = \frac{1}{2}$.
  • Substitute values: $\frac{ ext{root } 2}{2} imes \frac{ ext{root } 3}{2} + \frac{ ext{root } 2}{2} imes \frac{1}{2} = \frac{ ext{root } 6}{4} + \frac{ ext{root } 2}{4} = \frac{ ext{root } 6 + ext{root } 2}{4}$.

Example 4: Finding Exact Values of Sine in Radians

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

Example 5: Finding Tan Based on Sum of Angles

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

Example 6: Using Trigonometric Identities

• Problem: Given $ext{cos}(a) = \frac{2}{3}$ and $ext{sine}(b) = \frac{24}{25}$, 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 $ext{cos}(x + \frac{3 ext{pi}}{2}) = ext{sine}(x)$.
  • 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.