What is the Difference Between Sin 2x and 2 Sin x?

The difference between $$\sin^2(x)$$ and $$(\sin(x))^2$$ lies in the way the sine function is applied.

• $$\sin^2(x)$$: This expression represents the value of the sine function applied to $$x$$ and then multiplied by itself. In other words, it is the square of the sine function.
• $$(\sin(x))^2$$: This expression represents the value of the sine function applied to $$x$$, and then the result is squared. It is another way of writing the square of the sine function.

These two expressions are equivalent and represent the same mathematical concept. However, the notation $$\sin^2(x)$$ is often used to save space and simplify notation, while $$(\sin(x))^2$$ is more commonly used in textbooks and mathematical equations for clarity.

Comparative Table: Sin 2x vs 2 Sin x

The difference between $$\sin^2 x$$ and $$\sin(2x)$$ can be understood through their definitions and notations. Here is a comparison table:

• $$\sin^2 x$$ is a continuous function that maps an input angle $$x$$ to its square sine value. It has a domain of $$[-\pi, \pi]$$ and a range of $$[0, 1]$$. The graph of $$\sin^2 x$$ is U-shaped.

• $$\sin(2x)$$ is a continuous function that maps an input angle $$x$$ to its sine value after being doubled. It has a domain of $$[-\pi/2, \pi/2]$$ and a range of $$[-1, 1]$$. The graph of $$\sin(2x)$$ is V-shaped.

In summary, the main differences between $$\sin^2 x$$ and $$\sin(2x)$$ are their definitions, domains, ranges, and graph shapes.