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The half-angle formula for sine is a trigonometric identity that expresses the sine of half an angle in terms of the cosine of the original angle. It is derived from the double-angle formulas and is useful in various mathematical applications.
The calculator uses the half-angle formula:
Where:
Explanation: The formula calculates the sine of half an angle using the cosine of the original angle. The result is always non-negative for angles in the range where this identity is valid.
Details: Half-angle formulas are essential in trigonometry for simplifying expressions, solving equations, and calculating exact values of trigonometric functions for angles not commonly found on the unit circle.
Tips: Enter the cosine value (cos A) of the angle. The value must be between -1 and 1 inclusive. The calculator will compute the sine of half the angle.
Q1: Why is there a square root in the formula?
A: The square root comes from solving the double-angle formula for cosine: \( \cos A = 1 - 2\sin^2(A/2) \), which rearranges to \( \sin(A/2) = \sqrt{(1 - \cos A)/2} \).
Q2: When is the negative root used?
A: The sign depends on the quadrant of A/2. This calculator uses the principal (non-negative) square root. For complete solutions, the quadrant of A/2 must be considered.
Q3: What is the range of possible results?
A: The result sin(A/2) will always be between 0 and 1 inclusive, since we're taking the non-negative square root of a non-negative expression.
Q4: Can I use this for any angle?
A: The formula works for all real angles, but the calculator is limited to cosine values between -1 and 1, which covers all possible angles.
Q5: How accurate are the results?
A: The calculator provides results with high precision (up to 12 decimal places), making it suitable for most mathematical and engineering applications.