Sine Function
Sine function of an angle is a trigonometric function. The ratio of the lengths of the side opposite to the angle and the hypotenuse of a rightangled triangle is called the sine function which varies as the angle varies. It is defined in the context of a rightangled triangle for acute angles. Sine function is used to represent sound and light waves in the field of physics.
Sine function is simply denoted as sin x, where x is the angle. In this article, we will learn the basic properties of the sine function, its graph, domain and range, derivative, integral, and its power series expansion. The sine function is a periodic function and has a period of 2π.
1.  What Is a Sine Function? 
2.  Sine Function Graph 
3.  Values of Sine Function For Specific Angles 
4.  Properties of Sine Function 
5.  Identities of Sine Function 
6.  FAQs on Sine Function 
What Is a Sine Function?
The sine function is a periodic function in trigonometry. Consider a unit circle centered at the origin of the coordinate plane. A variable point P moves on the circumference of this circle. From the figure, we observe that P is in the first quadrant, and OP makes an acute angle of x radians with the positive xaxis. PQ is the perpendicular dropped from P onto the horizontal axis. The triangle is thus formed by joining the points O, P, and Q as shown in the figure.
Hence, the sine function for the above case can be mathematically written as:
sin x = PQ/OP, Here, x is the acute angle formed between the hypotenuse and the base of a rightangled triangle.
Sine Function Formula
The sine function can be defined as the ratio of the length of the perpendicular to the length of the hypotenuse in a rightangled triangle. Mathematically, the sine function formula in terms of sides of a rightangled triangle is written as:
sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse
Sine Function Graph
As shown in the image above, we note that sin x = PQ/OP = PQ/1 = PQ. As x varies, the value of the sine function sin x varies with the variation in the length of PQ. Now, we will study the variation in the sine function in the four quadrant of the coordinate plane.
Case 1: Variation of PQ in the first quadrant.
Suppose that initially, P is on the horizontal axis. Let us consider a movement of P through 90° or π/2 rad. The following figure shows different positions of P for this movement. Clearly, PQ has increased in length, from an initial value of 0 (when x is 0 radians) to a final value of 1 (when x is π/2 radians).
Case 2: Variation of PQ in the second quadrant.
Now, we will check the position of P in the second quadrant as we did in the first quadrant and check how the value of the sine function varies. P subsequently moves from 90° position to 180° position. In this phase of the movement, the length PQ decreases, from a maximum of 1 at 90°, to a minimum of 0 at 180°.
Case 3: Variation of PQ in the third quadrant.
When P moves from a position of 180° to a position of 270°, though the length or magnitude of PQ increases. But since the direction is along the negative yaxis, the actual value of sin x decreases from 0 to  1. Thus, the value of the sine function for angle x decreases.
Case 4: Variation of PQ in the fourth quadrant.
Finally, when P moves from a position of 270° to a position of 360°, sin x increases from −1 to 0 (once again). Though the length or magnitude of PQ decreases but the algebraic value of PQ will increase because its direction is along the negative yaxis. Thus, the value of the sine function for angle x increases.
We can now plot this variation on a graph. The horizontal axis represents the input variable x as the angle in radians, and the vertical axis represents the value of the sine function for x. Merging the response of variation in the value of PQ for all four quadrants, we obtained the complete plot of PQ vs x or sin x vs x, for one complete cycle of 0 radians to 2π radians (0° to 360°). The plot thus obtained is shown below:
Values of Sine Function For Specific Angles
We study the value of the sine function for some specific angles as they are easy to remember. These values are used in solving different mathematical problems. Some of these values of the sine function are listed below in the trigonometric table:
Sine Degrees  Sine Radians  Value of Sine Function (sin x) 

sin 0°  sin 0  0 
sin 30°  sin π/6  1/2 
sin 45°  sin π/4  1/√2 
sin 60°  sin π/3  √3/2 
sin 90°  sin π/2  1 
sin 120°  sin 2π/3  √3/2 
sin 150°  sin 5π/6  1/2 
sin 180°  sin π  0 
sin 270°  sin 3π/2  1 
sin 360°  sin 2π  0 
Properties of Sine Function
Properties of sine function depend upon the quadrant in which the angle lies. Sine function is a special trigonometric function and has many properties. Some of them are listed below:
 The sine graph repeats itself after 2π, which suggests the function is periodic with a period of 2π.
 The sine function is an odd function because sin(−x) = −sin x.
 The domain of sine function is all real numbers and the range is [1,1].
 The reciprocal of the sine function is the cosecant function.
 Power series expansion of the sine function is sin x = \(\sum_{n=0}^{\infty}(1)^n\dfrac{x^{2n+1}}{(2n+1)!}\)
Identities of Sine Function
In trigonometry, there are several identities involving the sine function. These identities are very useful in solving various math problems. Some of them are listed below:
 sin x = 1/ cosec x
 Inverse of sine function = sin^{1}x = arcsin x, where x lies in [1, 1]
 sin^{2}x + cos^{2}x = 1
 sin (x + y) = sin x cos y + sin y cos x
 sin (x  y) = sin x cos y  sin y cos x
 sin 2x = 2 sin x cos x
 Derivative of sine function: d(sin x)/dx = cos x
 Integral of sine function: ∫sin x dx = cos x + C, where C is the constant of integration
Related Topics on Sine Function
Important Notes on Sine Function
 Sine Function can be mathematically written as:
sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse  Sine Function is a periodic function with a period of 2π.
 The domain of sine function is (−∞, ∞) and the range is [−1,1].
Sine Function Examples

Example 1: Determine the value of the length of the perpendicular of a rightangled triangle if sin x = 0.6 and the length of the hypotenuse is 5 units using sine function formula.
Solution: We know that sin x = Perpendicular/Hypotenuse
We have sin x = 0.6, Hypotenuse = 5 units
Therefore, 0.6 = Perpendicular/5
⇒ Perpendicular = 5 × 0.6 = 3
Answer: Hence the length of the perpendicular is 3 units.

Example 2: Jennie was working on a construction site. Jenny wants to reach the top of the wall. A 44 ft long ladder connects a point on the ground to the top of the wall. The ladder makes an angle of 60 degrees with the ground. Can you find the height of the wall?
Solution: Given angle x = 60 degrees, Hypotenuse = 44 ft
⇒ sin 60° = √3/2
Using the sine function definition, we have sin x = Perpendicular/Hypotenuse
⇒ √3/2 = Perpendicular/44
⇒ Perpendicular = 22√3
Answer: The height of the wall is 22√3 ft.
FAQs on Sine Function
What is a Sine Function in Trigonometry?
Sine function of an angle is a trigonometric function. The ratio of the lengths of the side opposite to the angle and the hypotenuse of a rightangled triangle is called the sine function which varies as the angle varies.
What are the Properties of Sine Function?
Some of the properties of the sine function are:
 The sine graph repeats itself after 2π, which suggests the function is periodic with a period of 2π.
 The sine function is an odd function because sin(−x) = −sin x.
 The domain of sine function is all real numbers and the range is [1,1].
 The reciprocal of the sine function is the cosecant function.
 Power series expansion of the sine function is sin x = \(\sum_{n=0}^{\infty}(1)^n\dfrac{x^{2n+1}}{(2n+1)!}\)
What is the Definition of Sine Function?
The sine function can be defined as the ratio of the length of the perpendicular to the length of the hypotenuse in a rightangled triangle.
What is the Inverse Trigonometric Function of Sine Function?
Inverse of sine function = sin^{1}x = arcsin x, where x lies in [1, 1]
How do you Write a Sine Function?
A sine function can be written as sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse
What does the Graph of Sine Function Look Like?
The sine function curve is an updown curve that repeats after every 2π radians.
What is the Period of Sine Function?
A period of a function is when the function has a specific horizontal shift, P, results in a function equal to the original function, i.e., f(x+P) = f(x) for all values of x within the domain of f. The period of the sine function is 2π.
Is the Sine Function Even or Odd?
The sine function is an odd function because sin(−x) = −sin x.