The exact value of sin 180 is zero. Sine is one of the primary trigonometric functions which helps in determining the angle or sides of a right-angled triangle. It is also called trigonometric ratio. If theta is an angle, 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 0.

In Mathematics, trigonometry is the study of measurements of triangles that deals with the length, height and angles of the triangle. It has an enormous application in diverse fields like Science, Technology, Satellite navigations, and so on to calculate the varying measurements using cosine and sine function. In this article, the value of sin 180 degrees or the value of sin pi is discussed in detail.

What is the Value of Sine Pi (180°)?

Sin 180 is also denoted as sin pi or sin π in radians.

Note: Sin 180° = Sin 0° = 0

Sin 180 – Theta

One interesting fact related to Sin 180 degrees is sin 180 minus theta is equal to sin theta, where theta is any angle.

Sin (180° – Theta) = Sin Theta

sin (180° – θ) = sin θ

Hence, there are three points that we can conclude here:

• 180° – θ will come in IInd quadrant
• For any angle subtracted from 180°, sin will not be changed to cos
• The sign of sine ratio in the second quadrant will remain positive

How to derive the value of sin 180 Degrees?

The value of sin 180 (sin pi) can be interpreted in terms of different angles like 0°, 90° and 270°. Assume that the unit circle in the Cartesian plane is subdivided into four quadrants. And we know that the value of sin 180 degrees in the Cartesian plane takes place in the second quadrant. Therefore, the value of sin pi in the second quadrant is always positive.

From the value of sin 0, we will obtain the value of sin 180.

We know that the exact value of sin 0 degree is 0.

So, Sin 180 degree is +(sin 0) which is equal to +(0)

Therefore, the value of sin 180 degrees = 0.

The value of sin pi can be derived from some other trigonometric angles and functions like sine and cosine functions from the trigonometry table.

It is known that,

180° – 0° = 180° ———– (1)

270° – 90° = 180°———— (2)

Sine 180 Degree Derivation: Method 1

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

From the supplementary angle identity, we know that;

Sin (180 – Theta) = Sin Theta

Sin A = Sin (180° – A)

Therefore,

Sin ( 180° – A ) = Sin A

Sin ( 180° – 0° ) = Sin 0°

Sin 180° = 0 [Since the value Sin 0° is 0]

Hence, the value of sin pi is 0

Sine Pi Value Derivation: Method 2

From the expression (2),

Using complementary angle identity,

Sin A = cos (90°- A)

we can write the above expression as:

Sin 180°= cos (90° – 180°)

Sin 180°= cos (-90°)

Now, use opposite angle identity cos(-A) = cos A

Sin 180°= cos 90°

Sin 180°= 0 [Since the value of cos 90 degrees is 0]

Therefore, the value of sin 180 is 0.

Sin 180° = 0

Sin 180 and Sin 0

The value of sin 180 and sin 0 is equal to zero. 

Since 180 degrees angle falls under the second quadrant, therefore, the value of sine theta above 90 degrees changes to cosine function. That means;

Sin (90+A) = cos A

We can write, sin 180 as;

Sin 180 = sin (90+90) = cos 90

Since, we know, cos 90 = 0, therefore,

Sin 180 = 0

Also, we have known, sin 0 = 0

Therefore, sin 180 = sin 0.

The trigonometric ratios value for different angles and functions are as follows: