Question

What is \[\sin \left( -\theta \right)\] in terms of \[\sin \theta \]?

Answer 1

Hint: From the question we are asked to write \[\sin \left( -\theta \right)\] in terms of \[\sin \theta \]. For solving this question we will use the quadrant system in the trigonometry. We will discuss where the trigonometric function \[\sin \] is positive and where it is negative in the quadrant system and we will solve this question. So, we proceed with our solution as follows.

Complete step by step solution:
Generally the figure of the quadrant system indicating positive functions in the quadrants will be as follows.

Here in the above figure we can see four quadrants.
Now we will understand the conditions for the trigonometric function that is in the following four quadrants where the function \[\sin \] is positive and in which quadrants it is negative.
From basic trigonometry we know that in the first quadrant all the trigonometric functions are positive.
But in the second quadrant we will have the trigonometric function \[\sin \] as positive and all the remaining functions as negative.
Whereas, in the third and the fourth quadrant the trigonometric function \[\sin \] will attain negative values.
So, from the question we are given \[\sin \left( -\theta \right)\]. Which means the angle is in the fourth quadrant.
For sine, it is negative in the fourth quadrant as we discussed earlier.
Therefore,
\[\Rightarrow \sin \left( -\theta \right)=\sin \theta \]

Note: Students should have good knowledge in the concept of trigonometry and its applications. Students should know that the given angle is in the fourth quadrant. Students should not get into a conclusion that the angle is in the second quadrant as it makes our solution wrong.