Pre-Calc Unit 7.2

Combining Sine and Cosine Functions

• The discussion centers on the manipulation of sine and cosine functions involving the addition and subtraction of angles.

• Combining Signs:

  • When adding angles, such as in the question posed, one ponders the implications for the signs involved.

  • It's possible to add angles (e.g., using the formula: $ext{sin}(A + B)$) or to subtract them (e.g., with: $ext{sin}(A - B)$).

• Angle Example:

  • Entire angle: 63 degrees

  • Smaller angle: 45 degrees

  • Calculation involves:

  • $63 + 45 = 108$ degrees.

  • A further mention corrects this and notes the angles react as simple numerical additions.

• Basic Formulas for Sine and Cosine:

  • Formulas involve:

  • Sine of the sum: $ext{sin}(A + B) = ext{sin}(A) ext{cos}(B) + ext{cos}(A) ext{sin}(B)$

  • Sine of the difference: $ext{sin}(A - B) = ext{sin}(A) ext{cos}(B) - ext{cos}(A) ext{sin}(B)$

  • Cosine of the sum: $ext{cos}(A + B) = ext{cos}(A) ext{cos}(B) - ext{sin}(A) ext{sin}(B)$

  • Cosine of the difference: $ext{cos}(A - B) = ext{cos}(A) ext{cos}(B) + ext{sin}(A) ext{sin}(B)$

  • Tangent of the sum: $ext{tan}(A + B) = rac{ ext{tan}(A) + ext{tan}(B)}{1 - ext{tan}(A) ext{tan}(B)}$

  • Important that all formulas involve rational functions or ratios of sine and cosine.

Using Reference Angles for Sine Calculations

• Common Angles:

  • Key angles to memorize:

    • 30 degrees ($ rac{oldsymbol{ ext{ ext{π}}}}{6}$ radians)

    • 45 degrees ($ rac{oldsymbol{ ext{ ext{π}}}}{4}$ radians)

    • 60 degrees ($ rac{oldsymbol{ ext{ ext{π}}}}{3}$ radians)

    • 90 degrees ($ rac{oldsymbol{ ext{ ext{π}}}}{2}$ radians)

  • Focus on these angles to simplify calculations.

• Example Calculation:

  • Finding $ext{sin}(15^{ ext{circ}})$:

  • Recognizing it can be derived from $45^{ ext{circ}} - 30^{ ext{circ}}$ leads to:

  • Using the difference formula:

  • $ext{sin}(15^{ ext{circ}}) = ext{sin}(45^{ ext{circ}}) ext{cos}(30^{ ext{circ}}) - ext{cos}(45^{ ext{circ}}) ext{sin}(30^{ ext{circ}})$.

  • From the unit circle:

  • $ext{sin}(45^{ ext{circ}}) = rac{\boldsymbol{ ext{ ext{√2}}}}{2}$

  • $ext{cos}(30^{ ext{circ}}) = rac{\boldsymbol{3√3}}{6}$

  • $ext{cos}(45^{ ext{circ}}) = rac{\boldsymbol{ ext{√2}}}{2}$

  • $ext{sin}(30^{ ext{circ}}) = rac{1}{2}$

  • Subsequently, substituting leads to:

    • $ext{sin}(15^{ ext{circ}}) = rac{\boldsymbol{ ext{√2}}}{2} rac{3√3}{6} - rac{\boldsymbol{ ext{√2}}}{2} rac{1}{2}$

    • Final calculation:

    • $rac{ ext{√6} - ext{√2}}{4}$.

    • Confirming equality with further radical checks.

Transition to Radians and Additional Examples

• Next Calculations in Radians:

  • Keen transition to radians; for instance: $ext{pi}/12$ corresponds to 15 degrees.

  • Recognition of the formula involving $ext{cos}$ as:

  • Using formula number 15:

  • $ext{cos}(A - B) = ext{cos}(A) ext{cos}(B) + ext{sin}(A) ext{sin}(B)$

  • Example with $ext{sin}$ values and manipulation is similar to demonstrated above.

Proving Cofunction Relationships

• Cofunction Formula:

  • Establishes properties of sine and cosine:

  • $ext{sin}(90 - x) = ext{cos}(x)$

  • Introduces a right triangle example, confirming complementary angles (e.g., 30 degrees vs. 60 degrees, or 45 degrees vs. 45 degrees).

• Derivation of the Cofunction Relationship:

  • Using known formulas:

    • $ext{sin}(a - b) = ext{sin}(a) ext{cos}(b) - ext{cos}(a) ext{sin}(b)$ emphasizing sine and cosine ratio interrelations.

  • Continues with the tangent formulas and introduces a detailed exploration of numerical angles yielding sine and cosine values in context.

• Final Notes for Students:

  • Emphasis on correctly simplifying expressions, maintaining radicals as dictated by problem requirements.

  • Further practice will lead to deeper understanding.