Value of Sin 180

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The exact value of sin 180 is zero. Sine is known to be one of the primary trigonometric functions which help in determining the angle or sides of a right-angled triangle. It is also called trigonometric ratio. If theta is an angle in a right-angled triangle, then sine theta is equal to the ratio of perpendicular and hypotenuse of the right triangle. To be noted the value of sin 0 is also equal to 0.

In Mathematics, Trigonometry is the study of measurements of triangles that deals with the length, height, and angles of the triangle. Trigonometry has an enormous application in various fields such as Technology, Science, Satellite navigations, and so on to calculate the varying measurements using cosine and sine function. In this article, we are going to discuss the value of sin 180 degrees, or the value of sin pi is discussed in detail.

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: 

                                                [Image to be added Soon]

                                             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”.

What are Trigonometric Ratios?

According to the trigonometric ratio in maths, there are three primary or basic trigonometric ratios also known as trigonometric identities.

Sine Function Formula

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

Hence, we can write the formula as: 

Sin α = a(opposite) /h(hypotenuse)

                                                [Image to be added Soon]

The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H). The side that is opposite to the angle θ is known as the opposite(O). And the side which lies next to the angle  is known as the Adjacent(A)

The Pythagoras theorem states that, 

In a Right-Angle Triangle, (Opposite)2+(Adjacent)2= (Hypotenuse)2

Sine Value Table

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

Sine Value Table

                            Sine Degree

                    Sine Function Values

                                sin 0

                                0

                              sin 30

                              1/2

                              sin 45

                            1/2

                              sin 60

                          3/2

                              sin 90

                                1

                              sin 120

                          3/2

                              sin 150

                              1/2

                            sin 180

                              0

                            sin 270

                              -1

                            sin 360

                              0


Apart From These Main Sin Values, There are Few More Values of Sine Function:

sin 1= 0.84                                                                          sin 2=0.91 

sin 5= -0.96                                                                         sin 10= -0.54 

sin 20= 0.91                                                                        sin 30= -0.99                     

sin 40= 0.75                                                                        sin 50= -0.26

sin 70= 0.77                                                                        sin 80= -0.99

sin 100= -0.50                                                                     sin 105= -0.97

sin 210= 0.47                                                                      sin 240= 0.95

sin 330= -0.13                                                                     sin 350= 0.95      

Sine 180 Degree Derivation

Method 1:

Now we can use the above expression (1) in terms of sine functions

From the supplementary angle identity,

Sin A = Sin (180° – A )

Therefore,

Sin ( 180° – A ) equals to Sin A

Sin ( 180° – 0° ) equals to Sin 0°

Sin 180° equals to 0 [ Since the value Sin 0° is 0]

Hence, the value of sin pi is 0

    Table Showing the Value of Each Ratio with respect to Different Angles ( Trigonometric Ratios of Standard Angles Table).

Angle

    0 Degrees

30

Degrees

45

Degrees


60

Degrees 

90

Degrees

180

Degrees

270

Degrees

360

Degrees

Sin 

    0

1/2

1/√2

√3/2

  1

0

-1

0

Cos 

    1

√3/2

1/√2

½

  0

-1

0

1

Tan 

    0

1/√3

 1

 √3

  ∞

0

0

Cot 

    ∞

  √3

 1

      1/√3

  0

0

Cosec 

    ∞

 2

√2

   2/√3

 1

-1

Sec 

    1

    2/√3

√2

 2

-1

1

Properties of Sine as Per Quadrants

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 the Trigonometry diagram are shown below:

 [Image to be added Soon]                                        

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

Questions to be Solved

Q1) Calculate sin(A) from the triangle given below.

                 [Image to be added Soon]

Solution: We know the formula of sin(A) = \[\frac{Opposite}{Hypotenuse}\]

In the given question, 

Opposite = 11

Hypotenuse = 61

Then, sin(A) = \[\frac{11}{61}\]

FAQ (Frequently Asked Questions)

Q1: What are Three Basic Trigonometric Ratios?

Answer: The three basic trigonometric ratios in Mathematics are as follows: sin, cos , and tan. Let’s list down the three basic trigonometric ratios:

Name

Abbreviation

Relationship

Sine

Sin

Sin (θ)= Opposite/Hypotenuse

Cosine

Cos

Cos (θ)= Adjacent/Hypotenuse

Tangent

Tan

Tan (θ)= Opposite/Adjacent

Q2 :What are the 6 Trigonometric Functions of 180?

Answer:

  • Sin180 degrees=0.

  • Cos180 degrees=−1.

  • Tan180 degrees=0.

  • Sec180 degrees=−1.

  • Cot180 degrees=∞

Q3: What is the Value of Cos 180 Degrees?

Answer: We know that the exact value of cos 0 degrees is 1. So, cos 180 degrees which can be written as -(cos 0) which is equal to -(1) Therefore, the value of cos 180 degrees = -1. It is also represented in terms of radians.

Q4: How to Find Sin 180 Value?

Answer: We can write, sin 180 as;

Sin 180 = Sin (90+90)

Since, sin (90+a) = cos a

So, sin (90+90) = cos 90

As we know cos 90 = 0, therefore,

Sin 180 = 0