Question

Find the value of the function $ \sin {330^ \circ } $

Answer 1

Hint: We have 6 trigonometric functions in Mathematics; they are Sine, Cosine, Tan, Cot, Cosec and Sec. All the functions are considered positive in the first quadrant, only sine and cosec are positive in the 2nd quadrant, only tan and cot are positive in the 3rd quadrant, only cosine and sec are positive in the 4th quadrant. $ \sin {330^ \circ } $ falls under the 4th quadrant. Using this information to find its value.

Complete step-by-step answer:
We are given to find the value of the function $ \sin {330^ \circ } $
In $ \sin {330^ \circ } $ , the angle is $ {330^ \circ } $ which is greater than $ {270^ \circ } $ and less than $ {360^ \circ } $ .
The first quadrant ranges from $ {0^ \circ } $ to $ {90^ \circ } $ ; the second quadrant ranges from $ {90^ \circ } $ to $ {180^ \circ } $ ; the third quadrant ranges from $ {180^ \circ } $ to $ {270^ \circ } $ and the fourth quadrant ranges from $ {270^ \circ } $ to $ {360^ \circ } $ .
As we can see the angle $ {330^ \circ } $ lies in the fourth quadrant, in which only cosine and secant are positive and sine is negative.
So the value of $ \sin {330^ \circ } $ will be negative.
 $ \Rightarrow \sin {330^ \circ } $ can also be written as $ \sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right) $ .
The value of $ \sin \left( {{{360}^ \circ } - \theta } \right) = - \sin \theta $
In the same way, the value of $ \sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right) = - \sin {30^ \circ } $
The value of $ \sin {30^ \circ } $ is $ \dfrac{1}{2} $
$\Rightarrow$ This means $ \sin \left( {{{330}^ \circ }} \right) = - \sin {30^ \circ } = - \dfrac{1}{2} $
So, the correct answer is “ $ - \dfrac{1}{2} $ ”.

Note: Remember that when we subtract or add degrees from x of $ \sin x,\cos x,\tan x,\cos ecx,cotx,secx $ and the x is $ {180^ \circ } $ or $ {360^ \circ } $ , then the trigonometric function will not change; it stays the same. But if we subtract or add degrees from x of $ \sin x,\cos x,\tan x,\cos ecx,cotx,secx $ and the x is $ {90^ \circ } $ or $ {180^ \circ } $ , then sin becomes cos, tan becomes cot, cosec becomes sec and vice-versa. While finding the values of trigonometric functions, be careful with their signs because misplacing positive with negative can result wrong. As in $ \sin {330^ \circ } $ , we subtracted the angle from $ {360^ \circ } $ so the trigonometric ratio did not change.