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Question: How do you find the value of the expression \(\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \si...

How do you find the value of the expression sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘?\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ ?

Explanation

Solution

We will use some trigonometric identities to find the value of the given expression. We will use the trigonometric identity sin⁡xcos⁡y−cos⁡xsin⁡y=sin⁡(x−y).\sin x\cos y-\cos x\sin y=\sin \left( x-y \right). Then we will use the identity sin⁡300∘=sin⁡(360−60)=−sin⁡60∘.\sin 300{}^\circ =\sin \left( 360-60 \right)=-\sin 60{}^\circ .

Complete step by step solution:
Let us consider the given trigonometric expression sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘.\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ .
We are asked to find the value of this trigonometric equation.
To find the value of this equation, we will use some familiar trigonometric identities.
We know the trigonometric identity given by sin⁡(x−y)=sin⁡xcos⁡y−cos⁡xsin⁡y.\sin \left( x-y \right)=\sin x\cos y-\cos x\sin y.
We can see that the given expression resembles the right-hand side of the above written identity.
When we compare the values, we will get x=330∘x=330{}^\circ and y=30∘.y=30{}^\circ .
So, we can equate the right-hand side of the above written trigonometric identity with the given expression by replacing the variables from the identity with the values in the expression.
So, we will get the right-hand side of the identity as sin⁡(330−30).\sin \left( 330-30 \right).
Therefore, we can equate the given expression with the above written expression to find the value of the expression.
We will get sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=sin⁡(330−30).\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =\sin \left( 330-30 \right).
We know that 330−30=300.330-30=300. So, the above obtained trigonometric equation will become sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=sin⁡300∘.\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =\sin 300{}^\circ .
We know that 300=360−60.300=360-60.
Therefore, we will get the above equation as sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=sin⁡(360−60).\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =\sin \left( 360-60 \right).
Also, we have the identity sin⁡(360−x)=−sin⁡x.\sin \left( 360-x \right)=-\sin x.
So, we will get sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=−sin⁡60∘.\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =-\sin 60{}^\circ .
We know that sin⁡60∘=32.\sin 60{}^\circ =\dfrac{\sqrt{3}}{2}.
Therefore, we will get sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=−32.\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =-\dfrac{\sqrt{3}}{2}.

Note: There is an alternative method to find the value of the given expression. We know that 360−30=330.360-30=330. Therefore, sin⁡330∘=sin⁡(360−30)=−sin⁡30∘.\sin 330{}^\circ =\sin \left( 360-30 \right)=-\sin 30{}^\circ . Similarly, we will get cos⁡330∘=cos⁡(360−30)=cos⁡30∘.\cos 330{}^\circ =\cos \left( 360-30 \right)=\cos 30{}^\circ . We know that sin⁡30∘=12\sin 30{}^\circ =\dfrac{1}{2} and cos⁡30∘=32.\cos 30{}^\circ =\dfrac{\sqrt{3}}{2}. We will apply these values in the given expression to get sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=−sin⁡30∘cos⁡30∘−cos⁡30∘sin⁡30∘.\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =-\sin 30{}^\circ \cos 30{}^\circ -\cos 30{}^\circ \sin 30{}^\circ . And we will get sin⁡330∘cos⁡30∘−cos⁡330∘sin⁡30∘=−12×32−32×12=−234=−32.\sin 330{}^\circ \cos 30{}^\circ -\cos 330{}^\circ \sin 30{}^\circ =-\dfrac{1}{2}\times \dfrac{\sqrt{3}}{2}-\dfrac{\sqrt{3}}{2}\times \dfrac{1}{2}=-2\dfrac{\sqrt{3}}{4}=-\dfrac{\sqrt{3}}{2}.