Sin 2x vs 2 Sin x
Functions are one of the most important classes of mathematical objects, which are extensively used in almost all subfields of mathematics. The sine function which is denoted as f(x) = sin x is a trigonometric function defined from the set of real numbers onto the interval [-1, 1] and is periodic with period 2ᴫ.
The basic definition of the sine of an acute angle is done using a right-angled triangle. The sine of the angle is equal to the ratio of the length of the side opposite to an angle to the length of the hypotenuse. This definition can be extended to all the angles using the identities sin (-x) = – sin x and sin (ᴫ + x) = – sin x and sin (2nᴫ + x) = sin x.
For the next two sections consider f(x) = sin x and g(x) = 2x.
What is Sin 2x?
Consider the composite function f o g given by f o g (x) = f (g(x)) = f(2x) = sin 2x. This function is quite similar to sin x with the domain as the set of real numbers and the range as the interval [-1, 1]. This function is periodic with the period ᴫ (as opposed to the period 2ᴫ of sin x). Sin 2x can be expanded by the identity Sin 2x = 2 sin x cos x too.
What is 2 Sin x?
Consider the composite function g o f given by g o f (x) = g (f(x)) = g (sin x) = 2 sin x. This is also a periodic function with the same period as sin x, but twice the amplitude of it since -1 ≤ sin x ≤ 1 implies -2 ≤ 2 sin x ≤ 2. Its domain is the set of real numbers and the range is the interval [-2, 2]
What is the difference between Sin 2x and 2 Sin x?
• Sin 2x is defined from the set of real numbers onto the interval [-1, 1], whereas 2Sin x is defined from the set of real numbers onto the interval [-2, 2].
• Sin 2x is periodic with period ᴫ but 2 Sin x is periodic with period 2ᴫ.
The sin of some angle is easy. Get a ladder of length one metre, lie it on the floor and tilt up the ladder by the angle. The height of the end is called the sine or abbreviated sin. For small angles 2 sin x is nearly the same as sin 2x only if you measure angles in radians. To get radians multiply your degrees by 2 pi /360. For larger angles it is silly to imagine they could be equal. For example twice the square of 3 is obviously not the square of twice 3. Why should anyone think they are?