Compound Angles

In trigonometry, compound angles are formed by combining two or more angles to create a new angle. Understanding the trigonometric relationships involving compound angles is essential for simplifying expressions and solving problems. Here are the key concepts and formulas related to compound angles:

1. Sum and Difference Formulas:

  • Sine Sum and Difference:
    • sin(α±β)=sin(α)cos(β)±cos(α)sin(β)
  • Cosine Sum and Difference:
    • cos(α±β)=cos(α)cos(β)sin(α)sin(β)
  • These formulas express the sine and cosine of the sum or difference of two angles.

2. Double-Angle Formulas:

  • Sine Double-Angle:
    • sin(2θ)=2sin(θ)cos(θ)
  • Cosine Double-Angle:
    • cos(2θ)=cos2(θ)sin2(θ)
    • cos(2θ)=2cos2(θ)1
    • cos(2θ)=12sin2(θ)
  • These formulas express the sine and cosine of double angles in terms of the original angle.

3.  Triple-Angle Formulas:

  • The triple-angle formulas are extensions of the double-angle formulas.
  • Sine Triple-Angle:
    • sin(3θ)=3sin(θ)4sin3(θ)
  • Cosine Triple-Angle:
    • cos(3θ)=4cos3(θ)3cos(θ)
  • These formulas provide expressions for the sine and cosine of three times an angle.

4. Tangent Sum and Difference:

  • tan(α±β)=tan(α)±tan(β)1tan(α)tan(β)
  • This formula expresses the tangent of the sum or difference of two angles in terms of the tangents of the individual angles.

5. Product-to-Sum Formulas:

  • Sine Product-to-Sum:
    • sin(α)sin(β)=12[cos(αβ)cos(α+β)]
  • Cosine Product-to-Sum:
    • cos(α)cos(β)=12[cos(αβ)+cos(α+β)]
  • Sine-Cosine Product-to-Sum:
    • sin(α)cos(β)=12[sin(α+β)+sin(αβ)]
  • These formulas allow you to express the product of trigonometric functions as a sum or difference.

6. Half-Angle Formulas:

  • Sine Half-Angle:
    • sin(θ2)=±1cos(θ)2
  • Cosine Half-Angle:
    • cos(θ2)=±1+cos(θ)2
  • These formulas express the sine and cosine of half-angles in terms of the original angle.

Example:

  • Simplify sin(75) using compound angle formulas:
    • sin(75)=sin(45+30)
    • Applying the sum formula: sin(75)=sin(45)cos(30)+cos(45)sin(30)