Half Angle Formula: Understanding Sin (C/2)

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The half-angle formulas are trigonometric identities that allow you to find the trigonometric functions of an angle that is half of another angle. These formulas are particularly useful when you know the trigonometric functions of an angle but need to find the functions of half that angle.

Understanding the Half Angle Formula for Sine

The half-angle formula for sine is:

  • sin(x/2) = ±√((1 - cos x) / 2)

Key Components:

  • x: The angle for which you know the cosine value.
  • x/2: Half of the angle x, for which you want to find the sine.
  • ±: The plus or minus sign indicates that the sign of sin(x/2) depends on the quadrant in which x/2 lies.

How to Determine the Sign

To decide whether to use the positive or negative square root, consider the quadrant in which x/2 lies:

  • If x/2 is in the first or second quadrant, sin(x/2) is positive.
  • If x/2 is in the third or fourth quadrant, sin(x/2) is negative.

Using the Formula

To use the half-angle formula for sine, follow these steps:

  1. Determine the value of cos x.
  2. Substitute cos x into the formula.
  3. Simplify the expression inside the square root.
  4. Take the square root.
  5. Determine the correct sign (+ or -) based on the quadrant of x/2.

Example

Find the value of sin(15°), given that 15° is half of 30°, and cos(30°) = √3/2.

  1. Identify x: x = 30°
  2. Apply the formula: sin(15°) = ±√((1 - cos 30°) / 2)
  3. Substitute the value of cos 30°: sin(15°) = ±√((1 - √3/2) / 2)
  4. Simplify: sin(15°) = ±√((2 - √3) / 4)
  5. Further simplification: sin(15°) = ±(√(2 - √3)) / 2
  6. Determine the sign: Since 15° is in the first quadrant, sin(15°) is positive.
  7. Final result: sin(15°) = (√(2 - √3)) / 2

Practical Applications

The half-angle formulas are useful in various contexts:

  • Simplifying Trigonometric Expressions: They help in rewriting complex expressions in a simpler form.
  • Solving Equations: They can be used to solve trigonometric equations.
  • Calculus: They are used in integration and differentiation of trigonometric functions.

Tips for Success

  • Memorize the Formula: Knowing the formula is crucial for quick application.
  • Understand Quadrant Signs: Always determine the correct sign based on the quadrant of x/2.
  • Practice: Work through various examples to become comfortable with the formula.

The half-angle formula for sine is a powerful tool in trigonometry. By understanding its components and how to apply it, you can solve a variety of problems and simplify complex expressions. Remember to consider the quadrant to determine the correct sign and practice regularly to master its application.