** Sine vs Arcsine
**

Sine is one of the basic trigonometric ratios. It is an inevitable mathematical entity you find in any mathematical theory from the high school level onwards. Just as the Sine gives a value for a given angle, the angle for a given value can also be calculated. Arcsin or Inverse Sin is that process.

**More about Sine**

Sin can be defined basically in the context of a right angled triangle. In its basic form as a ratio, it is defined as the length of the side opposite the angle considered (α) divided by the length of the hypotenuse. sin α= (length of the opposite side )/(length of the hypotenuse).

In a much broader sense, the sin can be defined as a function of an angle, where the magnitude of the angle is given in radians. It is the length of the vertical orthogonal projection of the radius of a unit circle. In modern mathematics, it is also defined using Taylor series, or as solutions to certain differential equations.

The sine function has a domain ranging from negative infinity to positive infinity of real numbers, with the set of real numbers as the codomain too. But the range of the sine function is between -1 and +1. Mathematically, for all α belonging to real numbers, sin α belongs to the interval [-1,+1];{ ∀ α∈R ,sin α ∈[-1,+1] . That is, sin: R→ [-1,+1]

Following identities hold for the sine function;

Sin (nπ±α) = ± sin α ; When n∈Z and sin (nπ±α) = ± cos α when n∈ 1/2 ,3/2 ,5/2 ,7/2 …… (Odd multiples of 1/2). The reciprocal of the sine function is defined cosecant, with the domain R-{0} and range R.

**More about Arcsine (Inverse Sine)**

Inverse sine is known as the arcsine. In the inverse sine function, the angle is calculated for a given real number. In the inverse function, the relationship between the domain and the codomain is mapped backwards. The domain of the sine acts as the codomain for the arcsine, and the codomain for the sine acts as the domain. It’s a mapping of a real number from [-1,+1] to R

However, one problem with the inverse trigonometric functions is that their inverse is not valid for the whole domain of the considered original function. (Because it violates the definition of a function). Therefore, the range of the inverse sin is restricted to [-π,+π] so the elements in the domain are not mapped into multiple elements in the codomain. So sin^{-1}: [-1,+1]→ [-π,+π]

**What is the difference between Sine and Inverse Sine (Arcsine)?**

• Sine is a basic trigonometric function, and the arcsine is the inverse function of the sine.

• Sine function maps any real number/ angle in radians into a value between -1 and +1, whereas the arcsine maps a real number in [-1,+1] To [-π,+π]

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