1. What is the Half Angle Sine Formula?

The half-angle sine formula is a trigonometric identity that expresses the sine of half an angle in terms of the cosine of the original angle. It's derived from the cosine double-angle formula and is useful for simplifying trigonometric expressions.

2. How Does the Calculator Work?

The calculator uses the half-angle sine formula:

Where:

• \( \theta \) — Angle in radians
• \( \cos(\theta) \) — Cosine of the angle

Explanation: The formula calculates the sine of half the given angle by taking the square root of one minus the cosine of the angle, divided by two.

3. Importance of Half Angle Formulas

Details: Half-angle formulas are essential in trigonometry for simplifying complex expressions, solving integrals, and analyzing periodic phenomena in physics and engineering.

4. Using the Calculator

Tips: Enter the angle in radians. For degrees, first convert to radians (degrees × π/180). The result is dimensionless as it's a trigonometric ratio.

5. Frequently Asked Questions (FAQ)

Q1: Can I use degrees instead of radians?
A: The formula requires radians. Convert degrees to radians first by multiplying by π/180.

Q2: What's the range of valid results?
A: The result will always be between -1 and 1, like all sine values.

Q3: How accurate is this calculation?
A: The calculation uses PHP's built-in cosine function which is highly accurate for standard applications.

Q4: Are there other half-angle formulas?
A: Yes, there are similar formulas for cosine and tangent of half angles.

Q5: When would I use this in real applications?
A: Common uses include signal processing, vibration analysis, and solving trigonometric equations.