1. What is the Half Angle Formula for Sine?

Definition: The half-angle formula for sine relates the sine of half an angle to the cosine of the original angle.

Purpose: It allows calculation of trigonometric functions for angles that are half of a given angle, useful in trigonometry, calculus, and physics.

2. How Does the Formula Work?

The formula is:

Where:

• \( \theta \) — Original angle (in degrees or radians)
• \( \sin(\theta/2) \) — Sine of half the angle (dimensionless)
• The ± sign depends on the quadrant of θ/2

Explanation: The formula is derived from the cosine double-angle identity and provides an exact value for sine of half-angles.

3. Importance of Half Angle Formulas

Details: Half-angle formulas are essential for:

• Simplifying trigonometric expressions
• Solving trigonometric equations
• Calculating exact values for non-standard angles
• Applications in calculus, physics, and engineering

4. Using the Calculator

Tips:

• Enter the angle θ in degrees (between -360° and 360°)
• The calculator automatically determines the correct sign based on the quadrant
• Results are accurate to 6 decimal places

5. Frequently Asked Questions (FAQ)

Q1: Why is there a ± sign in the formula?
A: The sign depends on which quadrant θ/2 lies in. Our calculator automatically determines the correct sign.

Q2: Can I use radians instead of degrees?
A: This calculator uses degrees, but the formula works with radians too (1 radian ≈ 57.2958°).

Q3: What's the range of valid inputs?
A: The calculator accepts angles from -360° to 360°. Beyond this range, consider using angle reduction.

Q4: How accurate are the results?
A: Results are accurate to 6 decimal places, sufficient for most practical applications.

Q5: Are there similar formulas for cosine and tangent?
A: Yes! There are similar half-angle formulas for cosine and tangent functions.