Sine Function

Sine And Its Function

In trigonometry, there are a total of six trigonometric functions: sine, cos, tangent, secant, cosecant and cotangent. Out of all these six trigonometric functions, three are considered as primary functions and sine function is one of them. The rest two are tan and cos. We usually define sine theta as the ratio of the opposite side of the right angled triangle to its hypotenuse. Considering a triangle with ABC as an angle alpha, the sine function will be: 

                                               

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Sin α= Opposite/ Hypotenuse


Now we all know how confusing it is to remember the ratios of trigonometric functions but hey, we have got you a technique or rather a trick to make the remembering part easy and interesting. You can remember the trigonometric functions with the mnemonic SOH-CAH-TOA.

Where, SOH stands for “sine is opposite over hypotenuse”, CAH stands for “cosine is adjacent over hypotenuse” and TOA stands for “tangent is over adjacent”.

Sine Function Formula

The diagram below shows that Sin α = BC/AB.

Hence, we can write the formula as: 

Sin α = a/h


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Given below is the sine table from 0 degree to 360 degree with their respective values.

                                                             


Sine Value Table

                            Sine Degree

                    Sine function values

                                sin \[0^{o}\]

                                0

                              sin \[30^{o}\]

                            \[\frac{1}{2}\]

                              sin \[45^{o}\]

                      \[\frac{1}{\sqrt{2}}\]

                              sin \[60^{o}\]

                        \[\frac{\sqrt{3}}{2}\]

                              sin \[90^{0}\]

                                1

                              sin \[120^{o}\]

                        \[\frac{\sqrt{3}}{2}\]

                              sin \[150^{o}\]

                        \[\frac{1}{2}\]

                            sin \[180^{o}\]

                              0 

                            sin \[270^{o}\]

                              -1

                            sin \[360^{o}\]

                              0



Apart from these main since values, there are few more values of sine function:

sin \[1^{o}\] = 0.84                                                                          sin \[2^{o}\]=0.91 

sin \[5^{o}\]= -0.96                                                                          sin \[10^{o}\] = -0.54 

sin \[20^{o}\] = 0.91                                                                        sin \[30^{o}\]= -0.99                     

sin \[40^{o}\]= 0.75                                                                         sin \[50^{o}\] = -0.26

sin \[70^{o}\] = 0.77                                                                        sin \[80^{o}\] = -0.99

sin \[100^{o}\] = -0.50                                                                    sin \[105^{o}\] = -0.97

sin \[210^{o}\] = 0.47                                                                      sin \[240^{o}\]= 0.95

sin \[330^{o}\]= -0.13                                                                     sin \[350^{o}\] = 0.95      


Sine Identities

There are 5 common sine identities:

  1. sine (θ) = cos(π/2 − θ) = 1/cosec(θ)

  2. arcsin (sin θ) = θ, for −π/2 ≤ θ ≤ π/2

  3. cos2 (θ) + sin2 (θ) = 1

  4. Sin (2x) = 2sin (x) cos (x)

  5. Cos (2x) = cos2(x) − sin2(x)

The Other Trigonometric Functions are:

Trigonometric Functions

Representation As Sine

cos

\[\pm\] \[\sqrt{1 - sin^{2} (\theta}\]

tan

\[\pm\] \[\frac{sin(\theta)}{\sqrt{1 - sin^{2} (\theta)}}\]

Cot

\[\pm\] \frac{\sqrt{1 - sin^{2}(\theta)}}{sin(\theta)}

Sec

\[\pm\] \[\frac{1}{\sqrt{1- sin^{2}(\theta)}}\]

 

Trigonometry Law of Sines 

Trigonometry  law of sines established a relation between the sides a, b, and c and also the angles opposite to those sides A, B and C for an arbitrary triangle. Here are the relations: 

 

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We can see in the diagram above that A,B,C are the angles and a,b,c are the length of the sides. So according to the law of sine:


\[\frac{a}{sin A}\] = \[\frac{b}{sin B}\] = \[\frac{c}{sin C}\] = d

d = diameter of the triangle’s circumcircle. 

Sine Function Graph

The sine function graph, also called sine curve graph or a sinusoidal graph is an upside down graph. It repeats after every 360 at 2π

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Inverse Sine 

Inverse sine is also known as arcsine is a function which helps to measure the angle of a right angle triangle. It can also be denoted as asin or sin-1. 

Now, consider a right angle triangle with sides 1, 2, and √3. What we have to do is to calculate the angle a. So the sine function can be used as:


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Sin (a) = opposite/hypotenuse i.e ½.

We can calculate angle A using arcsine function. 

sin−1(½) = a

And therefore we get the value of angle “a” as 30°

Sine Calculus

According to the sine function f(x) = sin(x):

we can draw the integral of sin(x), ∫f(x) dx = −cos(x) + C (C= constant of integration) and derivative of sin(x),f′(x) as cos(x).


We can determine the values of sine function as positive or negative depending upon the quadrants. Here is a table where we can see that on one hand sine 270 is negative and on the other hand sine 90 is positive. Basically, for the first and the second quadrant it is positive and for the third and the fourth quadrant it is negative.  

The four quadrants in Trigonometry diagram are shown below:


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Properties of Sine as per Quadrants

Range of the degrees

      Quadrant

Sign of the sine function

Range of sin value

0-90

1st quadrant

Positive 

0 < sin(x) < 1

90-180

2nd quadrant

Positive 

0 < sin(x) < 1

180-270

3rd quadrant

Negative

-1 < sin(x) < 0

270-360

4th quadrant

Negative

-1 < sin(x) < 0


Solved Example:

Example 1: A ship is tethered to an anchor which is 30 meters long and creates a line that is the hypotenuse of a right triangle. It makes an angle between the hypotenuse and the ocean floor that is 39 degrees. So, what will be the depth of the anchor?

 

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Solution 1: It is practically not possible to dive all the way down to measure the depth of the anchor so we will use the trigonometric ratio to figure it out. Knowing the angle and length of the cable, we can find the length of the side opposite the angle, that is d, therefore, we use the sine function.

sin 39 = d/30

0.63 = d/30

d = 18.9 m


FAQ (Frequently Asked Questions)

1. What is the use of the sine function?

We can use sin functions to denote the angles of the triangle and modeling periodic phenomena, such as waves, sound and light. It is also used to solve problems of distance and height.


2. What does sine wave mean?

A sine wave is a plot of repetitive change or motion with two parameters plotted as a graph where time is often on the horizontal axis. If we take an example of Sound waves, we must know that they consist of very quick changes in air pressure which your ear interprets as sounds. A plot of air pressure can be graphed against time which would show them to be sine waves. In such audio tones, the pressure cycle hundreds or thousands of times per second.

3. Why sine is called sine? Sine is equal to what?

The term Sine is derived from a Latin word named sinus. This is a term that is related to a curve, or a fold, or a hollow. It is mostly interpreted as a fold of a garment, which was used as we would use a pocket today. In mathematics, it probably comes about through an incorrect translation of a Sanskrit word.  Sine is always equal to its opposite divided by the hypotenuse.