TRIG 8.3

The Double-Angle and Half-Angle Formulas

Objective 1: Understanding the Double-Angle Formulas

  • The Double-Angle Formulas express trigonometric functions of double angles in terms of single angles.

  • Formulas:

    • For sine:

    • extsin(2heta)=2extsin(heta)extcos(heta)ext{sin}(2 heta) = 2 ext{sin}( heta) ext{cos}( heta)

    • For cosine:

    • extcos(2heta)=extcos2(heta)−extsin2(heta)ext{cos}(2 heta) = ext{cos}^2( heta) - ext{sin}^2( heta)

    • This can also be expressed in equivalent forms:

      • extcos(2heta)=2extcos2(heta)−1ext{cos}(2 heta) = 2 ext{cos}^2( heta) - 1

      • extcos(2heta)=1−2extsin2(heta)ext{cos}(2 heta) = 1 - 2 ext{sin}^2( heta)

    • For tangent:

    • exttan(2heta)=rac2exttan(heta)1−exttan2(heta)ext{tan}(2 heta) = rac{2 ext{tan}( heta)}{1 - ext{tan}^2( heta)}

Objective 3: Understanding the Half-Angle Formulas

  • The Half-Angle Formulas express trigonometric functions of half angles in terms of functions of the full angle.

  • Formulas:

    • For sine:

    • Quadrant I or Quadrant II:

      • ext{sin}igg( rac{ heta}{2}igg) = ext{sqrt}igg( rac{1 - ext{cos}( heta)}{2}igg)

    • Quadrant III or Quadrant IV:

      • ext{sin}igg( rac{ heta}{2}igg) = - ext{sqrt}igg( rac{1 - ext{cos}( heta)}{2}igg)

    • For cosine:

    • Quadrant I or Quadrant IV:

      • ext{cos}igg( rac{ heta}{2}igg) = ext{sqrt}igg( rac{1 + ext{cos}( heta)}{2}igg)

    • Quadrant II or Quadrant III:

      • ext{cos}igg( rac{ heta}{2}igg) = - ext{sqrt}igg( rac{1 + ext{cos}( heta)}{2}igg)

    • For tangent:

    • General Form:

      • ext{tan}igg( rac{ heta}{2}igg) = rac{1 - ext{cos}( heta)}{ ext{sin}( heta)}

      • Alternative forms are:

      • ext{tan}igg( rac{ heta}{2}igg) = rac{ ext{sin}( heta)}{1 + ext{cos}( heta)}

      • ext{tan}igg( rac{ heta}{2}igg) = rac{1 + ext{cos}( heta)}{ ext{sin}( heta)}

Objective 5: Using the Double-Angle and Half-Angle Formulas to Evaluate Expressions Involving Inverse Trigonometric Functions

  • Application of these formulas aids in simplifying complicated trigonometric expressions and is particularly useful in integration and solving trigonometric equations.

  • May involve utilizing inverse functions to derive correct angle measures for given sine, cosine, or tangent values.

  • The ability to manipulate and derive other trigonometric functions using these foundational formulas is critical in calculus and analytical geometry.