Tan 3A Calculator

Tan 3A Formula:

\[ \tan 3A = \frac{(3 \times \tan A) - \tan^3 A}{1 - (3 \times \tan^2 A)} \]

Tan 3A:

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1. What is the Tan 3A Formula?

The Tan 3A formula is a trigonometric identity that expresses the tangent of three times an angle A in terms of the tangent of the angle A itself. It is derived from the triple-angle formulas in trigonometry.

2. How Does the Calculator Work?

The calculator uses the Tan 3A formula:

\[ \tan 3A = \frac{(3 \times \tan A) - \tan^3 A}{1 - (3 \times \tan^2 A)} \]

Where:

\( \tan A \) — The tangent value of angle A
\( \tan^2 A \) — Square of the tangent value
\( \tan^3 A \) — Cube of the tangent value

Explanation: This formula allows calculation of the tangent of triple an angle directly from the tangent of the original angle, without needing to know the angle measurement itself.

3. Trigonometric Identity Explanation

Details: The Tan 3A identity is derived from the sum formulas of trigonometry and is particularly useful in solving trigonometric equations and simplifying expressions involving triple angles.

4. Using the Calculator

Tips: Enter the value of tan A (tangent of angle A). The calculator will compute tan 3A using the trigonometric identity formula. Note that the result will be undefined if the denominator equals zero.

5. Frequently Asked Questions (FAQ)

Q1: When is the Tan 3A formula undefined?
A: The formula is undefined when the denominator equals zero, which occurs when \( 1 - 3\tan^2 A = 0 \), or \( \tan A = \pm \frac{1}{\sqrt{3}} \).

Q2: Can this formula be used for any angle?
A: Yes, the formula works for any angle A where both tan A and tan 3A are defined, except when the denominator becomes zero.

Q3: How is this formula derived?
A: The formula is derived using the tangent addition formula: \( \tan(3A) = \tan(2A + A) = \frac{\tan 2A + \tan A}{1 - \tan 2A \tan A} \), then substituting the double-angle formula for tan 2A.

Q4: What are the practical applications of this formula?
A: This formula is used in trigonometry problems, calculus, physics (especially wave mechanics), and engineering applications involving periodic functions.

Q5: Can I use this calculator for negative tan A values?
A: Yes, the calculator accepts both positive and negative tan A values and will compute the corresponding tan 3A accordingly.