Trigonometry: Compound Angles Formula

Compound angles are angles formed by adding or subtracting two or more angles. Understanding and applying the formulas for trigonometric functions of compound angles is crucial for simplifying complex expressions, solving trigonometric equations, and furthering your understanding of trigonometric relationships. This comprehensive guide provides a clear and concise overview of compound angle formulas, empowering you to expand your trigonometric toolkit.

What are Compound Angles?

Compound angles are expressions like A + B, A - B, 2A, 3A, etc., where A and B represent angles. The compound angle formulas express trigonometric functions of these combined angles in terms of trigonometric functions of the individual angles A and B.

Essential Compound Angle Formulas:

Here's a breakdown of the key compound angle formulas you'll need:

• Sine Formulas:

  • sin(A + B) = sin A cos B + cos A sin B
  • sin(A - B) = sin A cos B - cos A sin B

• Cosine Formulas:

  • cos(A + B) = cos A cos B - sin A sin B
  • cos(A - B) = cos A cos B + sin A sin B

• Tangent Formulas:

  • tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
  • tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Deriving Other Useful Formulas:

From these basic formulas, you can derive other useful identities:

• Double Angle Formulas (Special cases of compound angle formulas where A = B):

  • sin 2A = 2 sin A cos A
  • cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
  • tan 2A = 2 tan A / (1 - tan²A)

• Half Angle Formulas: These can be derived from the double angle formulas.

Illustrative Examples: Putting Formulas into Practice

Let's work through some examples to understand how to apply these formulas:

• Example 1: Find sin 75°. (Hint: 75° = 45° + 30°)

  • sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1) / 2√2

• Example 2: Find cos 15°. (Hint: 15° = 45° - 30°)

  • cos 15° = cos(45° - 30°) = cos 45° cos 30° + sin 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1) / 2√2

• Example 3: If tan A = 3/4 and tan B = 1/2, find tan(A + B).

  • tan(A + B) = (3/4 + 1/2) / (1 - (3/4)(1/2)) = (5/4) / (5/8) = 2

• Example 4: If sin A = 3/5 and cos B = 5/13, where A and B are acute angles, find sin(A+B).

  • First find cos A = √(1 - sin²A) = 4/5 and sin B = √(1 - cos²B) = 12/13.
  • Then sin(A+B) = sinAcosB + cosAsinB = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65

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Applications of Compound Angle Formulas:

Compound angle formulas are essential for:

• Simplifying complex trigonometric expressions.
• Solving trigonometric equations.
• Deriving other trigonometric identities.
• Calculations in physics, engineering, and other fields involving wave phenomena, oscillations, and rotations.

Conclusion: Expanding Your Trigonometric Abilities

Mastering compound angle formulas is a significant step in your trigonometric journey. This guide provides a valuable resource for learning and applying these essential formulas. By understanding and practicing with these formulas, you'll be well-equipped to tackle more advanced trigonometric problems and gain a deeper understanding of trigonometric relationships.

Call to Action:

Bookmark this page for quick reference and share it with others who might find it helpful. Practice applying these formulas to a variety of problems to reinforce your understanding. The more you work with these concepts, the more confident you'll become in using compound angle formulas.