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Question: Find the value of \[\sin ( - 210^\circ )\]....

Find the value of sin⁡(−210∘)\sin ( - 210^\circ ).

Explanation

Solution

Here we have to find the value of sine function at the angle of −210∘- 210^\circ. To do this first we have to find the value of sine function for negative angles. Then we write this angle in terms of the angles whose value is known easily to us. Then we have to locate the quadrant on the Cartesian plane where the angle 210∘210^\circ is located. Then we solve it further till we find the value of sin⁡(−210∘)\sin ( - 210^\circ ).

Formula used: We have used the following formulas here to solve the given question,
sin(180+θ)=−sinθsin(180 + \theta ) = - sin\theta
sin⁡(30∘)=12\sin (30^\circ ) = \dfrac{1}{2}
sin⁡(−θ)=−sin⁡θ\sin ( - \theta ) = - \sin \theta

Complete step by step solution:
We are given here to find the value of sin⁡(−210∘)\sin ( - 210^\circ ). Since we know that sin⁡(−θ)=−sin⁡θ\sin ( - \theta ) = - \sin \theta , we use this formula in our given question to move forward as,
sin⁡(−210∘)=−sin⁡(210∘)\sin ( - 210^\circ ) = - \sin (210^\circ )
Since, we can write 210∘210^\circ as (180∘+30∘)(180^\circ + 30^\circ ), we replace 210∘210^\circ in the above step with (180^\circ + 30^\circ )$$$$ \Rightarrow \sin ( - 210^\circ ) = - \sin (180^\circ + 30^\circ )
Since given angle is third quadrant, sign of sine function will be negative and we also know that, sin(180+θ)=−sinθsin(180 + \theta ) = - sin\theta
Using this, we get,

⇒sin⁡(−210∘)=−(−sin⁡(30∘)) ⇒sin⁡(−210∘)=sin⁡(30∘)  \Rightarrow \sin ( - 210^\circ ) = - ( - \sin (30^\circ )) \\\ \Rightarrow \sin ( - 210^\circ ) = \sin (30^\circ ) \\\

As, sin⁡(30∘)=12\sin (30^\circ ) = \dfrac{1}{2}, we put this value in above step and move ahead as,
⇒sin⁡(−210∘)=12\Rightarrow \sin ( - 210^\circ ) = \dfrac{1}{2}
Hence the value of the given function sin⁡(−210∘)\sin ( - 210^\circ ) comes out to be 12\dfrac{1}{2}.

Note:
While finding the values of the various trigonometric functions for the angles which are not the common angles which we study normally like 0∘,30∘,45∘,60∘,90∘,180∘,270∘,360∘0^\circ ,30^\circ ,45^\circ ,60^\circ ,90^\circ ,180^\circ ,270^\circ ,360^\circ , we try to break down or convert them in terms of these angles. We should be careful in locating the angle in quadrants as any mistake may lead to change in sign of the answer.