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Question: Find the value of the function \( \sin {330^ \circ } \)...

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

Explanation

Solution

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∘\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∘\sin {330^ \circ }
In sin⁡330∘\sin {330^ \circ } , the angle is 330∘{330^ \circ } which is greater than 270∘{270^ \circ } and less than 360∘{360^ \circ } .
The first quadrant ranges from 0∘{0^ \circ } to 90∘{90^ \circ } ; the second quadrant ranges from 90∘{90^ \circ } to 180∘{180^ \circ } ; the third quadrant ranges from 180∘{180^ \circ } to 270∘{270^ \circ } and the fourth quadrant ranges from 270∘{270^ \circ } to 360∘{360^ \circ } .
As we can see the angle 330∘{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∘\sin {330^ \circ } will be negative.
⇒sin⁡330∘\Rightarrow \sin {330^ \circ } can also be written as sin⁡(360∘−30∘)\sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right) .
The value of sin⁡(360∘−θ)=−sin⁡θ\sin \left( {{{360}^ \circ } - \theta } \right) = - \sin \theta
In the same way, the value of sin⁡(360∘−30∘)=−sin⁡30∘\sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right) = - \sin {30^ \circ }
The value of sin⁡30∘\sin {30^ \circ } is 12\dfrac{1}{2}
⇒\Rightarrow This means sin⁡(330∘)=−sin⁡30∘=−12\sin \left( {{{330}^ \circ }} \right) = - \sin {30^ \circ } = - \dfrac{1}{2}
So, the correct answer is “ −12- \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\sin x,\cos x,\tan x,\cos ecx,cotx,secx and the x is 180∘{180^ \circ } or 360∘{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\sin x,\cos x,\tan x,\cos ecx,cotx,secx and the x is 90∘{90^ \circ } or 180∘{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∘\sin {330^ \circ } , we subtracted the angle from 360∘{360^ \circ } so the trigonometric ratio did not change.