Tan 2A Formula – Proof and Examples

Here you will learn what is the formula of tan 2A in terms of tan with proof and examples based on it.

Let’s begin –

Tan 2A Formula :

The formula of tan 2A is \(2 tan A\over 1 – tan^2 A\)

Proof :

We have,

tan (A + B) = \(tan A + tan B\over 1 – tan A tan B\)

Replacing B by A,

\(\implies\)  tan 2A = \(tan A + tan A\over 1 – tan A tan A\)

\(\implies\)  tan 2A = \(2 tan A\over 1 – tan^2 A\)

We can also write above relation in terms of angle A/2, just replace A by A/2, we get

tan 2A = \(2 tan ({A\over 2})\over 1 – tan^2 ({A\over 2})\)

Example : Find the value of Tan 120 Degrees ?

Solution : We Know that tan 60 = \(\sqrt{3}\).

By using above formula, tan 2A = \(2 tan A\over 1 – tan^2 A\)

tan 120 = \(2 tan 60\over 1 – tan^2 60\) = \(2 \times \sqrt{3}\over 1 – 3\)

\(\implies\)  tan 120 = \(-\sqrt{3}\)

Example : If sin A = \(3\over 5\), where 0 < A < 90 degrees, find the value of tan 2A ?

Solution : We have,

sin A = \(3\over 5\) where 0 < A < 90 degrees

\(\therefore\) \(cos^2 A\) = 1 – \(sin^2 A\)

\(\implies\) cos A = \(\sqrt{1 – sin^2 A}\) = \(\sqrt{1 – {9\over 25}}\) = \(4\over 5\)

\(\implies\) tan A = \(sin A\over cos A\) = \(3/5\over 4/5\) = \(3\over 4\)

By using above formula,

tan 2A = \(2 tan A\over 1 – tan^2 A\) = \(2 \times {3\over 4} \over 1 – {9\over 16}\)

\(\implies\) tan 2A = \({6\over 4}\over {7\over 16}\)

\(\implies\)  tan 2A = \(24\over 7\)

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