Math Doubts

Sin double angle formula in terms of tan of angle

Formula

$\sin{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$

The sin double angle identity is usually expanded in terms of sin and cos of angle. It can also be used to expand in terms of tan of angle. For example, $\sin{(2θ)}$ is expanded in terms of $\tan{θ}$.

Other form

The sin double angle formula is usually written in several ways in which the following three are other popular forms.

$(1)\,\,\,\,\,\,$ $\sin{2x}$ $\,=\,$ $\dfrac{2\tan{x}}{1+\tan^2{x}}$

$(2)\,\,\,\,\,\,$ $\sin{2A}$ $\,=\,$ $\dfrac{2\tan{A}}{1+\tan^2{A}}$

$(3)\,\,\,\,\,\,$ $\sin{2\alpha}$ $\,=\,$ $\dfrac{2\tan{\alpha}}{1+\tan^2{\alpha}}$

The $\sin{(2x)}$ is expanded in terms of $\tan{x}$, the $\sin{(2A)}$ is expanded in terms of $\tan{A}$ and the $\sin{(2\alpha)}$ is expanded in terms of $\tan{\alpha}$. In the same way, any sin double angle function can be expanded in terms of the tan of respective angle.

Proof

Now, learn the proof to know how sin double angle formula is expanded in terms of tan of angle in trigonometry mathematics.



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