Question

Evaluate \[\sin {330^ \circ }\]?

Answer 1

Hint: Here in this question, we have to find the exact value of a given trigonometric function by using the sine sum or difference identity. First rewrite the given angle in the form of addition or difference, then the standard trigonometric formula sine sum i.e., \[\sin \left( {A + B} \right)\] or sine difference i.e., \[\sin \left( {A - B} \right)\] identity defined as \[\sin A \cdot \cos B + \cos A \cdot \sin B\] and \[\sin A \cdot \cos B - \cos A \cdot \sin B\] using one of these we get required value.

Complete step-by-step answer:
To evaluate the given question by using a formula of sine difference defined as the sine difference formula calculates the sine of an angle that is either the sum or difference of two other angles. It arises from the law of sines and the distance formula.
Consider the given function
\[ \Rightarrow \,\,\sin {330^ \circ }\]-------(1)
The angle \[{330^ \circ }\] can be written as \[{360^ \circ } - {30^ \circ }\], then
Equation (1) becomes
\[ \Rightarrow \,\sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right)\] ------(2)
Apply the trigonometric sine identity of difference \[\sin \left( {A - B} \right) = \sin A \cdot \cos B - \cos A \cdot \sin B\].
Here \[A = {360^ \circ }\] and \[B = {30^ \circ }\]
Substitute A and B in formula then
\[ \Rightarrow \,\,\sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right) = \sin {360^ \circ } \cdot \cos {30^ \circ } - \cos {360^ \circ } \cdot \sin {30^ \circ }\]
By using specified cosine and sine angle i.e., \[cos\,\,2\pi = cos\,36{0^ \circ } = 1\], \[cos\,\,\dfrac{\pi }{6} = \cos {30^0} = \dfrac{{\sqrt 3 }}{2}\], \[\sin \,2\pi = sin\,36{0^ \circ } = 0\] and \[\sin \,\,\dfrac{\pi }{6} = \sin {30^0} = \dfrac{1}{2}\].
On, Substituting the values, we have
\[ \Rightarrow \,\,\sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right) = 0 \cdot \dfrac{{\sqrt 3 }}{2} - 1 \cdot \dfrac{1}{2}\]
On simplification we get
\[ \Rightarrow \,\,\sin \left( {{{330}^ \circ }} \right) = - \dfrac{1}{2}\]
Hence, the exact functional value of \[\sin \left( {{{330}^ \circ }} \right) = - \dfrac{1}{2}\].

Note: Simply this can also be solve by using a ASTC rule i.e.,
\[ \Rightarrow \,\,\sin {330^ \circ } = \sin \left( {{{360}^ \circ } - {{30}^ \circ }} \right)\]
By using the ASTC rule of trigonometry, the angle \[{360^ \circ } - {30^ \circ }\] or angle \[2\pi - \dfrac{\pi }{6}\] lies in the fourth quadrant. sine function are negative here, hence the angle must in negative, then
\[ \Rightarrow \,\,\sin {330^ \circ } = - \sin \left( {{{30}^ \circ }} \right)\]
\[ \Rightarrow \,\,\sin \left( {{{330}^ \circ }} \right) = - \dfrac{1}{2}\]
While solving this type question, we must know about the ASTC rule.
And also know the cosine sum or difference identity, for this we have a standard formula. To find the value for the trigonometry function we need the table of trigonometry ratios for standard angles.