Trigonometry: Sub-Multiple Angles Formula

Sub-multiple angles, particularly half-angle formulas, are essential tools in trigonometry that allow us to express trigonometric functions of half an angle in terms of trigonometric functions of the full angle. These formulas are incredibly useful for simplifying expressions, solving equations, and expanding our understanding of trigonometric relationships. This comprehensive guide provides a clear and concise overview of sub-multiple angle formulas, empowering you to unlock new levels of trigonometric proficiency.

What are Sub-Multiple Angles?

A sub-multiple angle is a fraction of a given angle. The most commonly used sub-multiple angle formulas are the half-angle formulas, which express trigonometric functions of x/2 in terms of trigonometric functions of x.

Essential Sub-Multiple Angle Formulas (Half-Angle Formulas):

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

• Sine:

  • sin(x/2) = ±√((1 - cos x) / 2) (The sign depends on the quadrant of x/2)

• Cosine:

  • cos(x/2) = ±√((1 + cos x) / 2) (The sign depends on the quadrant of x/2)

• Tangent:

  • tan(x/2) = (1 - cos x) / sin x
  • tan(x/2) = sin x / (1 + cos x)
  • tan(x/2) = ±√((1 - cos x) / (1 + cos x)) (The sign depends on the quadrant of x/2)

Important Considerations for Half-Angle Formulas:

• Sign Determination: The ± sign in the sine and cosine half-angle formulas indicates that the correct sign (positive or negative) must be chosen based on the quadrant in which x/2 lies. Remember the ASTC rule (All Students Take Calculus) to determine the correct sign.

• Tangent Half-Angle: The tangent half-angle formula is often easier to use in its form without the square root, as it avoids the sign ambiguity.

Illustrative Examples: Putting Formulas into Practice

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

• Example 1: Find sin 15°. (Hint: 15° = 30°/2)

  • sin 15° = sin(30°/2) = √((1 - cos 30°) / 2) = √((1 - √3/2) / 2) = √((2 - √3) / 4) = (√(2 - √3)) / 2

• Example 2: Find cos 22.5°. (Hint: 22.5° = 45°/2)

  • cos 22.5° = cos(45°/2) = √((1 + cos 45°) / 2) = √((1 + 1/√2) / 2) = √((√2 + 1) / 2√2) = √((2+2√2)/4) = (√(2+2√2))/2

• Example 3: Find tan 67.5°. (Hint: 67.5° = 135°/2)

  • tan 67.5° = tan(135°/2) = (1 - cos 135°) / sin 135° = (1 - (-1/√2)) / (1/√2) = (√2 + 1) / 1 = √2 + 1

• Example 4: If cos x = -1/2 and x is in the second quadrant, find sin(x/2).

  • Since x is in the second quadrant (90° < x < 180°), x/2 will be in the first quadrant (45° < x/2 < 90°), so sin(x/2) will be positive.
  • sin(x/2) = √((1 - (-1/2)) / 2) = √(3/4) = √3 / 2

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Applications of Sub-Multiple Angle Formulas:

Sub-multiple angle formulas are used in:

• Simplifying trigonometric expressions and solving equations.
• Integration and differentiation of trigonometric functions.
• Geometric constructions and calculations.
• Physics and engineering applications involving wave phenomena and oscillations.

Conclusion: Expanding Your Trigonometric Horizons

Mastering sub-multiple angle formulas, particularly half-angle formulas, is a key step in developing your trigonometric skills. 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 a broader range of trigonometric problems and deepen your 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. Pay close attention to the signs of the trigonometric functions in different quadrants to ensure accuracy in your calculations.