Does sin²x = sin x²?

Many students have mistakenly think the "square of the sine of x" is the same as "the sine of the square of x."
This is one of the most common mistakes made in trigonometry … thinking that sin x² and sin² x are the same. They are not. They are very different.

• sin x² is asking you to take the sine of the square of some angle called x.
• sin² x is asking you to first take the sine of some angle called x … and to then square that ratio (the ratio of the side opposite to the hypotenuse of a right triangle with angle measure x).

For example, what if the angle measure is p/3?

and this does not equal

Additionally … notice how different the graphs of these two functions are.

y = sin x2	y = sin2 x

y = sin x²
This function has a range of [-1, 1] … because we square the angle before "taking its sine." Since squaring an angle will result in a new angle that is either 0 radians or some positive angle measure, the results of "taking the sine" of these angles will be something on the interval [-1, 1].

y = sin² x
This function has a range of [0, 1] … because we square only after "taking the sine" of the angles. Since the sine of any angle will be something on the interval [-1, 1], if we square these values, we will get only non-negative numbers that are on the interval [0, 1].

This is a classic case of following the order of operations. "Taking a trigonometric ratio of an angle" is a mathematics operation. The first step in the order of operations calls for doing all "operations" inside grouping symbols and students need to be aware that the trigonometric functions are implied grouping symbols.