Range of Sine

Range of trigonometric function Sine is defined as the set of values which are determined from Sine function for its domain. In other words, by substituting all angles belong to domain of sine; we can get all corresponding values and these values known as range.

Explanation of Range of Sine

Consider a right angle triangle and also assume angle between hypotenuse and adjacent side is Theta θ and we can write it as Sin θ. Similarly as per the definition of sine, It can express in mathematical form as stated below.

Sin θ

=

Opposite Side Hypotenuse

Similarly it can also further expand as per the Pythagorean Theorem.

Sin θ

=

Opposite Side √ [(Opposite Side)2 + (Adjacent Side)2]

If you adjust either angle or length of the opposite side, it gives corresponding value. For example, adjust angle of the right angle triangle to 30° and assume length of the opposite side and adjacent are 1 and √3 (≈ 1.732) respectively. Now just substitute these values in mathematical form of this function to get the required value.

Sin 30°

=

1 √ [(1)2 + (√3)2]

=

1 2

30° (Π/6) is an angle which is one of the values of the domain of sine function. Sine function gave us a value i.e ½ (0.5) for this angle but it is one of the values of its range. In this way, you can get all values for all angles. Even you can plot a graph by taking angles' values in horizontal axis and corresponding values in vertical axis.

You clearly observe that the maximum value of sine is 1 in both regions. In other words whenever the adjacent side is zero or angle reaches 90°, sine gives maximum value. For every 2Π interval, the same graph will be repeated. However, this graph does not cross the value 1 in both positive and negative regions. We can say, the minimum value of sine is -1 and the maximum value of sine is 1. Within -1 to 1 region, sine function gives values but we cannot write or show all values of sine between -1 to 1 in order to write range of this function.

However, range of the sine denoted with [-1, 1] which means all values within -1 to 1 and the reason of showing it with [ and ] brackets is we should also consider values -1 and 1. In other words denoting range with [ and ] brackets is it gives all vales from -1 to 1.