Sine Graph

Sin Graph is a graphical representation of a relation between angles of right angle triangle and sine values for the respective angles. Simply, it is a graph between various angles and sine values at those angles. In fact analyzing this graph helps us to know the properties of the trigonometric function Sine. Similarly, this graph is popularly used in physics and engineering applications most importantly signals and systems, electrical engineering and etc.

Consider a right angle triangle and its angle is Theta, denoted with θ symbol. The sides of the same right angle triangle are opposite side, adjacent side and hypotenuse as shown in the below image. By considering angle, trigonometric ratio sine can be written as Sin θ.

According to definition of Sine function, it is a ratio between length of the opposite side and hypotenuse of a right angle triangle and can write it in mathematical form as stated below.

Sin θ

=

Length of the Opposite Side Length of the Hypotenuse

Setting length of the opposite side to different values changes the angle of the right angle triangle and influences the length of the adjacent side as well as hypotenuse. You can determine all values for all angles including negative angles by using above mathematical form. Here are some examples for your understanding purpose.

Sin 0° = 0
Sin 15° = 0.2588
Sin 30° = 0.5
Sin 45° = 0.7071
Sin 60° = 0.866
Sin 90° = 1

Now plot a graph by considering all angles on horizontal axis and corresponding sine values on vertical axis and you finally see a graph as shown in the below animated presentation.

Once you carefully observe this graph, you can understand the properties of sine as well as its graph.

1. As the angle goes on increasing up to 90°, sine function travels towards its peak value but after 90°, it stars falling down towards zero and continues its journey into negative region after 180°. However, it reaches its minimum value at 270° and starts rising towards zero to complete its duty at 360°.
2. For every 360° interval the same path will be repeated. Hence, the periodicity (wave length) is 2Π.
3. This graph reaches its maximum value in positive region and minimum value in negative region. However, it reaches unit value i.e 1 in both regions. In magnitude point of view, sine graph minimum and maximum values are same but remember signs are different.
4. Sine graph never breaks up at any point and continues its journey by following the same path in both directions. That’s why the domain of sine is Real Numbers (R).
5. Similarly, this sine does not cross the unit value in both regions which means it reaches the -1 value in negative region and 1 in positive region for every period. Hence, the range of the sine is [-1, 1].