Formula and Execution

Angle α

α is the angle which you will substitute here, and from the dropdown arrow, select the unit of the angle which is given to you, and our online sine calculator will give instant results.

Sine Calculator

Are you looking for the values of angles of the sine? Whether your angle is in degrees, radians, or gradians, our online sine calculator makes it super easy for you to get the values in whatever units you need. Just enter your angle, select the measuring unit from our dropdown menu and bingo! You get instant, precise results before you even blink your eye!

Concept of Sine

Before moving on to the concept of sine, let us learn some trigonometry and know where sine originated from.

Trigonometry

Trigonometry has been in human use since the times when an actual term was not even formalized for it. Trigonometry has helped people travel across the oceans, to measure distances and heights for building structures, mountains peaks, and this does not limit here; its vastness was extended to the use in celestial bodies as well, that is how we got the first distance measure to the moon. Till the day, trigonometry is used in all of these areas and even more. The applications of trigonometry always have souped-up. The word trigonometry is derived from the Greek words ‘trigon’ meaning triangle and ‘metron’ meaning measurement.

Let us consider a triangle with vertices A, B, and C. The sides of the triangle are named as a, b, and c with respect to their vertex. The side opposite to vertex A is a, the side opposite to vertex B is b, and similarly, the side opposite to the vertex C is c.

When it comes to the study of triangles, what do you think we have to study in it? Three angles and three sides, of course!

In the triangle considered, our angles are <A, <B, and <C, and the lengths of the sides are a, b, and c, respectively. In trigonometry, we deal with these six things. A few are given to us while using trigonometry; we calculate others.

The thing is that trigonometry is fixated to the study of only right-angled triangles, also known as right triangles. And the good news is that our work already gets simpler when one angle is always known, and it is 90 degrees.

For now, we consider a right triangle with an angle <C = 90 degrees. The other two angles are named <A and <B, respectively.

Consider <A as the angle with respect to which we will derive our trigonometric ratios. The sides associated with this angle are the side adjacent to it means the side next to it which is side b, the side opposite to it which is side a, and the hypotenuse c which is always opposite to the right angle.

The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are abbreviated as sin, cos, tan, cot, sec, and csc, respectively.

Now look at the following formulas that define these trigonometric ratios.

Sin <A = opposite/hypotenuse = a/c

Cos <A = adjacent/hypotenuse = b/c

Tan <A = opposite/adjacent = a/b

The above three are the basic trigonometric ratios and they can be easily learned by the acronym SOH-CAH-TOA. Where the alphabets are the initials of the functions and sides. Like in SOH, S is sine, O is opposite, and H is the hypotenuse. In CAH, C is cos, A is adjacent, and H is hypotenuse. The same goes for TOA with T standing for tan, O for opposite, and A for adjacent.

One more thing to notice here is that tangent is the ratio of sine by cosine. When we divide the ratios of sine and cosine i.e., (opposite/hypotenuse)/ (adjacent/hypotenuse), we get opposite/adjacent which is equal to tangent ratio. So, we can say that,

tan <A = sin <A/cos <A

I talked about six trigonometric ratios on top. So, you must be thinking what about the other three? Well, the other three are extremely easy to remember. The cotangent or cot function is the reciprocal of tangent or tan function, similarly sec, and csc are the reciprocal of sin and cos functions respectively. So we define them in the following way:

Cot <A = 1/tan <A = adjacent/opposite = c/a

Sec <A = 1/cos <A = hypotenuse/adjacent = c/b

Csc <A = 1/sin <A = hypotenuse/opposite = c/a

These six functions have covered all the possible ratios of the sides of the right triangle.

Now, when the concept of trigonometry is clear, we will define the sine function as follows:

Definition of Sine Function

The sine function is the ratio of the side opposite to the angle considered to the hypotenuse in a right triangle. To study further about the sine function and its properties, we will introduce the concept of angles and the units to measure these angles.

Concept of Angle and its Measurement

An angle is the measure of the rotation of a given ray about its initial point. To explain this definition, we consider a ray OA and rotate it to an arbitrary position B. The new ray formed is known as OB. The ray OA is the initial ray and is known as the initial side while the ray OB is known as the terminal side. There is an angle subtended between the two rays and the common point O is known as the vertex.

We rotated the ray in the counterclockwise direction here. We can also rotate the ray in a clockwise direction. The difference would be the sign. The counterclockwise measurement of an angle is positive while the angle in the clockwise direction is negative.

Units of Measurement of Angles

Angles are measured in different units. A few of them are discussed below.

1. Degree

The degree is the most common unit to measure an angle. It is denoted by a small zero like symbol that is placed as the superscript of the number like 10. We know that one full rotation is equal to 360 degrees. So 1 degree would be equal to 1/360 degrees.

2. Minutes

Minutes is another unit used to measure angles. This is smaller than degrees and is represented by a prime like 1'.

1 degree = 60 minutes

Or 10 = 60'

3. Seconds

Seconds is smaller than minutes and double prime is used to represent second. 1 minute is equal to 60 seconds.

1' = 60"

This means that

10 = 60*60 = 360"

We use minutes and seconds as a measure of angles for more accuracy. Like 500 25' 4" is more accurate than just 500.

4. Radians

This is another important unit used to measure angles. To understand the concept of radians, consider a circle with radius r units. Radians is defined as the angle subtended at the center of the circle by an arc equal in length to the radius is 1 radian.

You can see in the figure. The radius of the circle is r units. The length of arc AB is also r units. So the angle AOB is equal to 1 radian.

Now consider another circle with radius r units and the distance travelled by the arc is equal to 2r units. So the angle would be equal to 2 radians. In general, we can say, if the length of the arc is s units and the radius is r units, then

x = s/r

Where x is the angle subtended by the radius. The size of the angle is given by the ratio of the arc length to the length of the radius. The other unit of measure of angles is gradians.

Relation Between Degrees and Radians

Consider a semi-circle with radius r and angle x. The length of arc s i.e., the outer boundary of the semi-circle, would be half the circumference of the circle i.e.,

s = 1/2 (2(pi)r)

s = (pi) r

 

And as we studied above,

x = s/r = (pi)r/r = pi

 

We know that the angle subtended in a semi-circle is 180 degrees. So,

(pi) rad = 180 degrees

 

From the above relation, we can deduce

pi/2 rad = 90 degrees

3pi/2 rad = 270 degrees

2pi rad = 360 degrees

pi/6 rad = 30 degrees

pi/4 rad = 45 degrees

pi/3 rad = 60 degrees

These are the most commonly used angles in trigonometry. You can deduce further angles by using the relation pi rad= 180 degrees.

Signs of Sine Function in Four Quadrants

The sign of the trigonometric function depends on the quadrant in which the angle lies.

Consider the following figure

The angle is in the first quadrant and the point P has coordinates (a, b). We know that sine is the ratio of the opposite by hypotenuse. So, from the figure we can write as

Sin x = b/r

Since the x and y coordinates are positive in the first quadrant, we get positive values of sine in this first quadrant, we get positive values of sine in this quadrant.

Now for the second quadrant, the coordinates of P are (-a, b), so the ratio of the sine of angle x would become

Sin x = b/r

In this quadrant, too, we get positive values of the sine of an angle.

In the third quadrant, the P coordinates are (-a, -b), we can write sine ratio as,

Sin x = -b/r

So the angle that lies in the third quadrant will have negative values of sine.

Similarly, for the fourth quadrant, the coordinates of P are (a, -b), the sine ratio becomes

Sin x = -b/r

The sine value of the angle in the fourth quadrant would, therefore, be negative.

So, we can summarize the result by saying that sine is positive in 1st and 2nd quadrants and it is negative in 3rd and 4th quadrants.

Values of Sine Function

QuadrantsAngle in DegreesAngle in RadiansExact ValueDecimal Value

 

 

1st Quadrant

00000
300pi/6½0.5
450pi/4(√2)/20.707
600pi/3(√3)/20.866
900pi/211

 

2nd Quadrant

12002pi/3(√3)/20.866
13503pi/4(√2)/20.707
15005pi/6½0.5
1800pi00

 

3rd Quadrant

21007pi/6-1/2-0.5
22505pi/4-(√2)/2-0.707
24004pi/3-(√3)/2-0.866
27003pi/2-1-1

 

4th Quadrant

30005pi/3-(√3)/2-0.866
31507pi/4-(√2)/2-0.707
330011pi/6-1/2-0.5
36002pi00

The above table shows the values of the sine function that are used frequently. Notice how the sign changes with respect to the quadrant.

Period of Sine Function

A periodic function is the one that repeats itself after the smallest positive value is added to the domain of the function. Sine function is periodic and it repeats the value of range after a period of 2pi radians or 360 degrees.

Example:

The value of sin 30 or sin (pi/6) is 1/2. If I add 360 degrees or 2pi radians to the angle i.e., sin (30+360) or sin (pi/6 + 2pi), we again get 1/2 as the result.

Graph of Sine Function

The graph of the sine function is a continuous wave that rolls gently and keeps repeating itself after the period of 2pi as we saw above. The graph of the sine function for the domain of -2pi to 2pi is shown below.

We see from the graph that the maximum value of sine is 1 when the angle is 90 degrees or pi/2 radians and the minimum value is -1 when the angle 270 degrees or 3pi/2 radians. Sine takes the value of 0 at 180 degrees and 360 degrees or pi radians and 2pi radians respectively.

FAQs

1. What is the domain and range of sine function?

The domain is the set of values we put in the function and the set of values we get as results are known as the range. We know that if the variable of a function appears in the denominator, it constraints the domain of the function because if the denominator becomes zero, the function becomes undefined. Look at the example to understand the concept.

 

Let f(x) = (x+2)/(x-3)

The above function gets undefined when the value of x is 3 because the denominator becomes zero at this value. So, we say that the domain of the above function is all real values except 3.

When we talk about the domain of sin x, the function does not have anything in the denominator to limit the set of values of the domain. So we say that the domain of sine function is all real values and we write it as (-inf, inf), where inf stands for infinity.

The range of the sine function is the set of values we get as results we substitute the domain in the function.

We know that sine is always the measure of the opposite side divided by the hypotenuse. Because the hypotenuse is always the longest side, the number on the bottom of the ratio will always be larger than that on the top. For this reason, the output of the sine function will always be a proper fraction. It will never be a number greater than or equal to 1 unless the opposite is equal in length to the hypotenuse (which happens only when your triangle is a single segment and you are working with circles – see cosine calculator).

So when we consider the angles in the counterclockwise direction, our values of the sine function would be positive and the maximum positive value possible is 1. And when go clockwise, our least possible value is -1 because the function generates negative values in this direction.

So we say that the range of sine function is between -1 to 1 and we write it as [-1, 1]

2. What are the properties of sine function?

Sine function exhibits very interesting and distinctive characteristics. Its properties are:

  • It is periodic with a period of 2(pi). Or more generally, we can say 2n(pi), where n is the set of integers.
  • It is an odd function i.e., sin(-x) = -sin(x)
  • The domain of sine function is all real numbers while the range is from -1 to 1.
  • The graph of the sine function is symmetric about the origin.
  • The y-intercept of the sine function occurs at 0.
  • Multiple x-intercepts exist because of periodic nature and they occur at the multiples of n(pi).

3. How do you calculate sine?

Sine is simply the ratio of the opposite side by the hypotenuse. To measure sine of the angle, consider the opposite side with respect to the angle and divide it by the hypotenuse to get the result.

Another simple way is to simply substitute the value of the angle in our online sine calculator to get quick results.

4. What is sin-1 on the calculator?

It is the inverse of sine i.e., it calculates the angle for you if the ratio opposite/hypotenuse is given. Consider the example

Sin(x) = ½

x = sin-1 (1/2)

x = 300

The ratio of sine function was given to us, and sin-1 helped us find the angle.

5. Is the inverse of sine i.e., sin-1 and the reciprocal of sine i.e., 1/sine the same thing?

No, they are not the same thing. The inverse of sine generates the value of angle for us when the ration opposite/hypotenuse is given to us. While reciprocal of sine is the ratio of sine considered inversely i.e., hypotenuse/opposite.