Sine

The sine of an angle is a trigonometric function that is denoted by sin x, where x is the angle in consideration. In a right-angled triangle, the ratio of the perpendicular and the hypotenuse is called the sine function. In other words, it is the ratio of the side opposite to the angle in consideration and the hypotenuse and its value vary as the angle varies. The sine function is used to represent sound and light waves in the field of physics.

In this article, we will learn the basic properties of sin x, sine graph, its domain and range, derivative, integral, and power series expansion. The sine function is a periodic function and has a period of 2π. We will solve a few examples using the sine function for a better understanding of the concept.

The sine of an angle in a right-angled triangle is a ratio of the side opposite to an angle and the hypotenuse. The sine function is an important periodic function in trigonometry and has a period of 2π. To understand the derivation of sin x, let us consider a unit circle centered at the origin of the coordinate plane. A variable point P moves on the boundary (circumference) of this circle. Observe that P is in the first quadrant, and OP makes an acute angle of x radians with the positive x-axis. 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 between the hypotenuse and the base of a right-angled triangle OPQ.

The sine of an angle can be defined as the ratio of the length of the perpendicular to the length of the hypotenuse in a right-angled triangle. It gives the value of the sine function of the angle between the base and hypotenuse of the right triangle. Mathematically, it is abbreviated as sin x, where x is an acute angle between the base and hypotenuse of the right triangle.

The sine function is written as the ratio of the length of the perpendicular and hypotenuse of the right-angled triangle. Mathematically, the sine function formula in terms of sides of a right-angled triangle is written as:

sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse

The domain of sine function is all real numbers as sin x is defined for all x in (-∞, ∞). Whereas the range of sin x is [-1, 1] as the value of sin x does not go beyond this. The graph of sine function looks like a wave that oscillates between -1 and 1. Also, the period of sin x is 2π as its value repeats after every 2π radians. The domain and range of sine function can also be observed using its graph. In the section below, let us see how the graph of sin x is made.

Before getting to the graph of the sine function, let us understand how the values of sine vary on a unit circle and then plot them on the graph. As shown in the image above, we note that sin x = PQ/OP = PQ/1 = PQ (As the radius of a unit circle is 1, so OP = 1). As x varies, the value of sin x varies with the variation in the length of PQ. Now, we will study the variation in the sine function in the four quadrants 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 of 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 y-axis, the actual value of sine function 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 magnitude value of PQ will increase because its direction is along the negative y-axis. Thus, the value of the sine 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. 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 sine graph thus obtained is shown below:

Now, let us go through the values of the sine function for some specific angles such as 0°, 30°, 45°, 60°, 90°, etc. as they are easy to remember. Most of the values given below are used for solving different problems in trigonometry. The values of sin x are listed below in the trigonometric table:

Properties of sine function depend upon the quadrant in which the angle lies. Sine is a special trigonometric function and has many properties. Some of them are listed below:

• The sine graph repeats itself after a period of 2π, which shows the function is periodic with a period of 2π.
• Power series expansion of the sine function is sin x = \(\sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n+1}}{(2n+1)!}\)
• The sine function is an odd function because sin(−x) = −sin x.
• The reciprocal of the sin x is the cosec x.
• The domain of the sine is all real numbers and the range is [-1,1].

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 is sin^-1x = arcsin x, where x lies in [-1, 1]
• sin²x + cos²x = 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 sin x: d(sin x)/dx = cos x
• Integral of sin x: ∫sin x dx = -cos x + C, where C is the constant of integration

☛ Related Topics:

• Cosine Function
• Inverse Trigonometric Ratios
• Trigonometric Ratios

Important Notes on Sine Function:

• Sine can be mathematically written as:
  sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse
• f(x) = sin x is a periodic function and sine function period is 2π.
• The domain and range of the sine are (−∞, ∞) and [−1,1], respectively.

What is a Sine in Trigonometry?

The sine of an angle is a trigonometric function, also known as the sine function. The ratio of the lengths of the side opposite to the angle and the hypotenuse of a right-angled triangle is called the sine function which varies as the angle varies and it is abbreviated as sin x, where x is an acute angle between the base and the hypotenuse.

How to Find the Period of Sine Function?

To find the period of sine function f(x) = Asin Bx + C, we use the formula, Period = 2π/|B|. For sine function f(x) = sin x, we have A = 1, B = 1 , C = 0. We substitute this value of B into the formula to find the sine function period. Hence, the period of sin x is given by, Period = 2π/|1| = 2π.

What is the Range of Sine Function?

The range of sine function is [-1, 1] as the graph of sin x oscillates between -1 and 1 only. The value of the sine function does not go beyond -1 and 1.

How to Find the Amplitude of a Sine Function?

The amplitude of the sine function f(x) = Asin Bx + C is given by the value A. For f(x) = sin x, we have A = 1, B = 1 , C = 0. Therefore, the amplitude of sine function sin x is equal to 1.

Is Sine Function Bijective?

Sine Function defined on all real numbers is not bijective. However, if we restrict the domain of sin x to [-π/2, π/2] and redefine the function as f(x) = sin x, f: [-π/2, π/2] → [-1, 1], then sine function becomes bijective.

What is Inverse Sine?

Inverse of sine function is given by sin^-1x = arcsin x, where x lies in [-1, 1]. Inverse of sine function is read as 'arc sine x' or 'sine inverse x'.

What is Sine Function Formula?

A sine function formula is given by sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse. This formula helps to determine the value of the sine function sin x which can be determined using the lengths of the sides of the triangle or using the angle x.

Is Sine Continuous?

The sine graph is continuous as there are no breaks or gaps in the sine curve and can be drawn without lifting the pen. Also, the sine function is defined for all real numbers, hence the graph does not break and can be drawn continuously. Hence, we can conclude that the sine function is continuous.

What is the Period of Sine?

The period of the sine function is 2π as the values of sin x repeat after every 2π radians. This can also be seen through the sine function graph as the values of sin x oscillate between -1 and 1 and repeat after every 2π radians. Hence, the sine function period is 2π.

Is Sine Function Even or Odd?

The sine function is an odd function it satisfies the definition of an odd function, that is, f(-x) = -f(x), for all x. We have f(-x) = sin(−x) = −sin x = -f(x). Hence, sin x is an odd function.

What is the Value of Sine?

The value of sine varies as the angle between the base and hypotenuse of a right-angled triangles changes. The commonly used values of the sine are: sin 0 = 0, sin π/6 = 1/2, sin π/4 = 1/√2, sin π/3 = √3/2, and sin π/2 = 1. We can determine these values using the sine formula given by, sin x = Perpendicular/Hypotenuse.