1. What is the Tan (A/2) Formula?

The tan(A/2) formula is a trigonometric identity that expresses the tangent of half an angle in terms of the cosine of the full angle. It is derived from the half-angle formulas in trigonometry and is useful for solving various mathematical problems.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( \cos A \) — Cosine of angle A (must be between -1 and 1)
• \( \tan(A/2) \) — Tangent of half angle A

Explanation: The formula calculates the tangent of half the given angle using the cosine value of the full angle. The square root ensures we get the positive root for the principal value.

3. Applications of Half-Angle Formula

Details: The half-angle formula is used in trigonometry, calculus, physics, and engineering for solving equations, integration problems, and analyzing periodic phenomena where half angles are involved.

4. Using the Calculator

Tips: Enter the cosine value of angle A (must be between -1 and 1). The calculator will compute the tangent of half the angle A. Ensure the input is valid to avoid mathematical errors.

5. Frequently Asked Questions (FAQ)

Q1: Why does the formula use square root?
A: The square root is part of the mathematical derivation of the half-angle formula from the double-angle identities in trigonometry.

Q2: What if the denominator is zero?
A: When cos A = -1, the denominator becomes zero, making the result undefined. This occurs when A = 180° + k·360° (or π + 2kπ radians).

Q3: Are there alternative forms of this formula?
A: Yes, tan(A/2) can also be expressed as (1 - cos A)/sin A or sin A/(1 + cos A), which are equivalent forms.

Q4: What is the range of possible results?
A: The result can be any real number, from negative infinity to positive infinity, depending on the value of cos A.

Q5: Can this formula be used for any angle?
A: The formula works for all angles except those where cos A = -1, as this would cause division by zero.