Find the value of \[\sin {{4530}^{o}}\].

- Hint: Convert the angle \[{{4530}^{o}}\] as multiple of \[{{180}^{o}}\], i.e. write \[4530\text{ as }n\pi +\theta \], where \[\pi \] is radian form of \[{{180}^{o}}\]. Now, get the quadrant in which \[{{4530}^{o}}\] is lying and change the function according to it. Identity related to this concept is given as:
\[\begin{align}
  & \sin \left( n\pi +\theta \right)=\sin \theta ,\text{ if n is even} \\
 & \sin \left( n\pi +\theta \right)=-\sin \theta ,\text{ if n is odd} \\
\end{align}\]

Complete step-by-step solution -

Let us assume the given value in the question as M.
\[M=\sin {{4530}^{o}}....\left( i \right)\]
As the angle \[{{4530}^{o}}\] is greater than \[{{90}^{o}}\] and we do not know the value of any trigonometric function at an angle greater than \[{{90}^{o}}\], so we need to use some trigonometric relation to convert the given expression to an expression, which involves acute angle with it. So, let us divide 4530 by 180 to write it in multiple of \[{{180}^{o}}\]. So, we get,
\[{{4530}^{o}}={{180}^{o}}\times 25+{{30}^{o}}....\left( ii \right)\]
As, we know the radian form of the angle \[{{180}^{o}}\] is given as \[{{180}^{o}}=\pi \], hence we get the radian form for the angle \[{{30}^{o}}\] as,
\[{{180}^{o}}=\pi \]
\[{{30}^{o}}=\dfrac{\pi }{{{180}^{o}}}\times {{30}^{o}}=\dfrac{\pi }{6}\]
\[{{30}^{o}}=\dfrac{\pi }{6}\]
Hence, equation (ii) can be represented as
\[{{4530}^{o}}=25\pi +\dfrac{\pi }{6}....\left( iii \right)\]
Hence, equation (i) can be expressed as
\[M=\sin \left( 25\pi +\dfrac{\pi }{6} \right)....\left( iv \right)\]
Now, we know the quadrants in trigonometry is given as

As \[25\pi \] is an odd multiple of \[\pi \], it means \[25\pi \] will lie at the exact position of \[\pi \] because if we rotate \[2\pi \] from \[\pi \] in the above diagram, we get the angle \[3\pi \] and if we rotate more by \[2\pi \], we get \[5\pi \] and hence, the angles have a generalized form at an angle \[\pi \], that are
\[\pi ,3\pi ,5\pi ,7\pi .....\left( 2n-1 \right)\pi \]
Hence, \[25\pi \] will also lie at the same position of \[\pi \]. Now, we are adding \[\dfrac{\pi }{6}\] to it in the equation (iv), i.e. the angle given is \[25\pi +\dfrac{\pi }{6}\]. Hence, the angle \[25\pi +\dfrac{\pi }{6}\] will lie in the third quadrant as \[25\pi \] is lying at \[\pi \] and we are adding \[\dfrac{\pi }{6}\] to it. Now, as we know sin function is negative in the third quadrant, and angle involved in the equation (iv) is an odd multiple of \[\pi \], so the trigonometric function will not change while conversion. So, we can write the identities related to these concepts as:
\[\sin \left( \pi +\theta \right)=-\sin \theta \]
\[\sin \left( 3\pi +\theta \right)=-\sin \theta \]
\[\sin \left( 5\pi +\theta \right)=-\sin \theta \]
….
\[\sin \left( n\pi +\theta \right)=-\sin \theta \]
where n is an odd integer. So, we can write equation (iv) as,
\[M=\sin \left( 25\pi +\dfrac{\pi }{4} \right)=-\sin \dfrac{\pi }{6}\]
Now, as we know the value of \[\sin \dfrac{\pi }{6}\] is given as \[\sin \dfrac{\pi }{6}=\dfrac{1}{2}\].
Hence, the value of M can be given as \[M=\dfrac{-1}{2}\].
So, the value of \[\sin {{4530}^{o}}\text{ is }\dfrac{-1}{2}\]. Hence, \[\dfrac{-1}{2}\] is the answer.

Note: We need to know two important rules involved for the conversion of trigonometric functions with respect to the angles.
(i) Take care of the sign with the help of the given trigonometric function and the quadrant in which the angle is lying.
(ii) If the angle involved inside the trigonometric function is multiple of \[\dfrac{\pi }{2}\], i.e. \[\dfrac{n\pi }{2}\pm \theta \] type, where n is an odd integer, then we need to convert sin to cos, tan to cot, sec to cosec and vice versa. And if the angle involved in the sum is multiple of \[\pi \], i.e. \[n\pi \pm \theta \] type, then the trigonometric function will remain the same.
Use the above two rules for the conversion of any trigonometric function by changing their angles. Do not go for direct calculation of the exact value with other identities, it will be a complex approach and may lead to wrong answers. So, always try to solve these kind of questions by the way given in the solution.