Why the value of sin function is always in between 1 and -1?

Answer 1

2

Answer 2

The answer will depends on how the function is defined.

The sine function is usually introduced in mathematics as a trigonometric ratio: the ratio of the opposite side and the hypotenuse in a right angled triangle. Thanks to Pythagoras, the hypotenuse must be the largest side in a triangle and that means that the absolute value of the sine function must be less than 1.

Alternatively, if you can establish the identity

sin2(A) + cos2(A) = 1 then it follows that these bounds apply.

An equivalent definition, in terms of coordinate geometry is sine(A) = y/sqrt(x2+y2) where the line joining P(x,y) to the origin makes an angle of A with the x axis. It is easy to show that sin(A) must therefore lie between -1 and 1. The negative values of the sine function occur when the angle in question is in the third or fourth quadrants so that y is negative.

The more advanced definition for the sine function is

sin(a) = a/1! - a3/3! + a5/5! - a7/7! + ... where the angle a is measured in radians. It can be shown that this function is equivalent to the trigonometric definition.