sin 2θ

Formula

$\sin 2 \theta = 2 \sin \theta \cos \theta$

Proof

sine two theta right angled triangle
Step: 1

According to $\Delta UOS$

The angle of the right angled triangle $UOS$ is $2\theta$. So, express sine of angle $2\theta$ in its ratio form.

$\sin 2 \theta = \dfrac{SU}{OS}$

The side $\overline{RW}$ perpendicularly intersects the side $\overline{SU}$. Therefore, the length of the side $\overline{SU}$ can be expressed as the sum of the lengths of the sides $\overline{SW}$ and $\overline{WU}$.

$\implies \sin 2 \theta = \dfrac{SW + WU}{OS}$

$\implies \sin 2 \theta = \dfrac{SW}{OS} + \dfrac{WU}{OS}$

The sides $\overline{WU}$ and $\overline{PR}$ are parallel lines and their lengths are absolutely same. Hence, the length of the side $\overline{WU}$ can be replaced by the length of the side $\overline{PR}$.

$\implies \sin 2 \theta = \dfrac{SW}{OS} + \dfrac{PR}{OS}$

Step: 2

According to $\Delta WSR$

right angled triangle wsr

Consider right angled triangle $WSR$ and express cosine of angle theta in terms of its ratio form.

$\cos \theta = \dfrac{SW}{SR}$

$\implies SW = SR \cos \theta$

Now replace the length of the side $\overline{SW}$ by this value in the sine two theta expression.

$\therefore \,\,\, \sin 2 \theta = \dfrac{SR \cos \theta}{OS} + \dfrac{PR}{OS}$

Step: 3

According to $\Delta POR$

right angled triangle por

Consider the right angled triangle $POR$ and write the sine of angle theta in terms of its ratio form.

$\sin \theta = \dfrac{PR}{OR}$

$\implies PR = OR \sin \theta$

Replace the length of the side $\overline{PR}$ by the value in the same sine two theta equation.

$\therefore \,\,\, \sin 2 \theta = \dfrac{SR \cos \theta}{OS} + \dfrac{OR \sin \theta}{OS}$

$\implies \sin 2 \theta = \dfrac{SR}{OS} \cos \theta + \dfrac{OR}{OS} \sin \theta$

Step: 4

According to $\Delta ROS$

right angled triangle ros

Finally, consider the right angled triangle $ROS$. Express the sine and cosine values in their respective ratio form.

$\sin \theta = \dfrac{SR}{OS}$

$\cos \theta = \dfrac{OR}{OS}$

Now, replace the ratio of lengths of the sides $\overline{SR}$ to $\overline{OS}$ by sin of angle theta and also replace the ratio of lengths of the sides $\overline{OR}$ to $\overline{OS}$ by cosine of angle theta, in sine of angle two theta equation.

$\implies \sin 2 \theta = \sin \theta \cos \theta + \cos \theta \sin \theta$

$\therefore \,\,\, \sin 2 \theta = 2 \sin \theta \cos \theta$

Therefore, It is proved that the sine of angle two theta is equal to twice the product of sine of angle theta and cosine of angle theta.

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