Question

What is the value of $ \sin {180^ \circ }? $
A. $ 1 $
B. $ 0 $
C. $ - 1 $
D. $ \dfrac{1}{2} $

Answer 1

Hint: In this question we have to find the value of $ \sin 180^\circ $ . We can see that this is a trigonometric question, as sine, cosine, and tangent are trigonometric ratios. They are also called the basic trigonometric functions.
 We can solve this question by using trigonometric identities, so we will use the formula
 $ \sin 2A = 2\sin A\cos A $ .
We know that the value of
 $ \cos 90^\circ = 0 $ And,
 $ \sin 90^\circ = 1 $ .

Complete step-by-step answer:
There are several methods by which we can calculate the value of $ \sin 180^\circ $ .
We can write $ \sin 180^\circ $ as
$ \sin (2 \times 90^\circ ) $
Now we can apply the formula here
$ \sin 2A = 2\sin A\cos A $ .
Here we have
$ A = 90 $
So we can write
 $ 2\sin 90^\circ \cos 90^\circ $
We know the value of
$ \cos 90^\circ = 0 $
And,
$ \sin 90^\circ = 1 $
By putting these values back in the equation we have:
$ 2 \times 1 \times 0 = 0 $
So it gives us value
 $ \sin 180^\circ = 0 $ .
There is another method by which we can derive the value.
We can write $ \sin 180^\circ $ as
$ \sin (90 + 90) $
Here we will apply another trigonometric formula
$ \sin (a + b) = \sin a\cos b + \cos a\sin b $
By comparing we have
 $ a = b = 90^\circ $
So we can write
 $ \sin 90^\circ \cos 90^\circ + \cos 90^\circ \sin 90^\circ $
We can put the value and it gives us:
 $ 1 \times 0 + 0 \times 1 = 0 $
Hence it gives us the answer $ \sin 180^\circ = 0 $ .
So the correct option is (B) $ 0 $ .
So, the correct answer is “Option B”.

Note: We should note that we can use the complementary identity also to solve this question:
 $ \sin A = \cos (90 - A) $ .
So we can write
 $ \sin 180^\circ $ as $ \cos (90^\circ - 180^\circ ) $
It gives us value
 $ \cos ( - 90^\circ ) $
We will now use the opposite angle identity i.e.
 $ \cos ( - A) = \cos A $
Here we have
 $ A = 90 $
So we can write
 $ \cos ( - 90) = \cos 90^\circ $
And we know the value of
 $ \cos 90^\circ = 0 $ .
Hence
 $ \sin 180^\circ = \cos 90^\circ = 0 $ .