Question

What is \[\sin \left( -{{90}^{\circ }} \right)\]?

Answer 1

Hint: We are given a trigonometric function with an angel and we have to find the value of the same. We know that sine function is an odd function. Also, the negative sign in the angle means that we will go clockwise and not anti-clockwise. So, we can have the function also as \[\sin \left( {{270}^{\circ }} \right)\]. Using all the information we discussed, we will have the value of the given trigonometric function.

Complete step by step solution:
According to the given question, we are given a trigonometric function with a specific and we have to find its value.
The given trigonometric expression we have is,
\[\sin \left( -{{90}^{\circ }} \right)\]----(1)
In the equation (1), we can see that the angle has a negative sign to it. It means we have to go clockwise and not anti-clockwise. We also know that the sine function has positive values in the first and the second quadrant only and in the third and fourth quadrant, the sine function has negative values.
Here, \[\sin \left( -{{90}^{\circ }} \right)\] can also be written as \[\sin \left( {{270}^{\circ }} \right)\]. The value of the sine function remains the same but when moved in different quadrants the signs of the sine values change. For example –
\[\sin \left( {{90}^{\circ }} \right)=1\] and when in the fourth quadrant , we have, \[\sin \left( {{270}^{\circ }} \right)=-1\].
So, the value for \[\sin \left( -{{90}^{\circ }} \right)=\sin \left( {{270}^{\circ }} \right)=-1\].

Therefore, the value of \[\sin \left( -{{90}^{\circ }} \right)=-1\].

Note: We can also solve the above question using the concept of odd and even function. We know that sine is an odd function, that is, \[f(-x)=-f(x)\].
So, we will have,
\[\sin \left( -{{90}^{\circ }} \right)=-\sin \left( {{90}^{\circ }} \right)=-1\]
Hence, the value of \[\sin \left( -{{90}^{\circ }} \right)=-1\].