Question

Is $\sin 2x$ the same as $2\sin x$?

Answer 1

Identifying is $\sin \left(2x\right)$ is the same as $2\sin x$:

$\sin \left(2x\right)$ is not the same as $2\sin \left(x\right)$.

Sine of twice of an angle $\left(x\right)$ is equal to twice of $\sin \left(x\right)\cos \left(x\right)$.

By trigonometric identities,

$\sin 2x=2\sin \left(x\right)\cos \left(x\right)...\left(1\right)$

where $x$is a reference angle in a right-angled triangle

Therefore,

$2\sin \left(x\right)=\frac{\sin \left(2x\right)}{\cos \left(x\right)}...\left(2\right)$

Hence, we can see from the above equation that $2\sin x$ is not equal to $2\sin x$.

We can also prove it by an example;

Suppose, if$x=\frac{\pi }{2}$, then,

$\sin \left(2x\right)=\sin 2\left(\frac{\pi }{2}\right)=\sin \pi =0...\left(3\right)$

Now,

$2\sin x=2\sin \left(\frac{\pi }{2}\right)=2\left(1\right)=2...\left(4\right)\mathbf{}\left[\because sin\frac{\pi }{2}=1\right]\mathbf{}$

From equations $\left(3\right)\text{and}\left(4\right)$, we get;

$\sin \left(2x\right)\ne 2\sin x$

Hence, it is evident that $\sin 2x$ is not the same as $2\sin x$.