1. What is the Tangent Double Angle Formula?

The tangent double angle formula relates the tangent of an angle to the tangent of twice that angle. It's derived from trigonometric identities and is useful for simplifying trigonometric expressions and solving equations.

2. How Does the Calculator Work?

The calculator uses the double angle formula:

Where:

• \( θ \) — The angle in degrees
• \( \tan(θ) \) — Tangent of the angle (optional input)
• \( \tan(2θ) \) — Tangent of the double angle

Explanation: The formula shows that the tangent of double an angle can be calculated from the tangent of the original angle. The result becomes undefined when tan²(θ) = 1 (at θ = 45° + k·90°).

3. Applications of Double Angle Formula

Details: The double angle formula is used in trigonometry, calculus, physics, and engineering for simplifying expressions, solving equations, and analyzing periodic phenomena.

4. Using the Calculator

Tips: Enter the angle in degrees. You may optionally provide the tangent of the angle if known (for higher precision). The calculator will compute tan(θ) if not provided.

5. Frequently Asked Questions (FAQ)

Q1: When is tan(2θ) undefined?
A: tan(2θ) is undefined when θ = 45° + k·90° (k integer), where the denominator becomes zero.

Q2: Can I use radians instead of degrees?
A: This calculator uses degrees. For radians, convert your angle to degrees first (180° = π radians).

Q3: Why would I provide tan(θ) instead of letting the calculator compute it?
A: For exact values (like tan(45°) = 1) or when you have a more precise measurement than standard floating-point calculation would provide.

Q4: What's the range of tan(2θ)?
A: tan(2θ) can be any real number, from negative to positive infinity, except at points where it's undefined.

Q5: Are there similar formulas for sin(2θ) and cos(2θ)?
A: Yes: sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ).