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# The value of $\sin {10^ \circ } + \sin {20^ \circ } + \sin {30^ \circ }........\sin {360^ \circ }$ is equal to

Last updated date: 16th Sep 2024
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Hint: Though this is an easy sum, it is very tricky and difficult if one is not well acquainted with the identities and values of Trigonometric functions. This sum involves use of sine properties . The property which we are going to use in this particular sum is $\sin (360 - \theta ) = - \sin \theta$ . We can also use the graph to understand this property or just memorize these types of expressions for all the trigonometric functions.

From the given sum we will have club expressions in which the sum of angle turns out to be ${360^ \circ }$
Thus we club $\sin {10^ \circ }\& \sin {350^ \circ }$ , $\sin {20^ \circ }\& \sin {340^ \circ }$ $\sin {30^ \circ }\& \sin {330^ \circ }$ till we reach ${360^ \circ }$ .
Total numbers of terms involved in the given expression are ${36^ \circ }$ .
If we club $2$ terms, then we will be having $18$ pairs.
In order to solve the sum we will use the property $\sin (360 - \theta ) = - \sin \theta$ . This will help us in finding the solution for each pair.
Now the last step is finding the value of each pair in order to reach the final answer.
Considering the $1st$ pair $\sin {350^ \circ }$
We can write $\sin {350^ \circ } = \sin ({360^ \circ } - {10^ \circ }).....(1)$
From Equation $1$ we can say that the value of $\sin {350^ \circ }$ is the same as $- \sin {10^ \circ }$ .
$\therefore$ Pair one which is $\sin {10^ \circ }\& \sin {350^ \circ }$ can now be written as $\sin {10^ \circ }\& - \sin {10^ \circ }$
Since the numerical involves the sum of series, the value of Pair 1 would become $0$ .
Similarly for all the pairs the value would be $0$ .
$\therefore$ The sum of the series would be $0$
So, the correct answer is “0”.

Note: This sum is just the application of the properties of trigonometric functions. If the student finds it difficult in memorizing the properties it is advisable to learn them by making graphs. Graphical representation is another method of understanding these properties. Numericals and word problems from the chapter of Trigonometry would be only based on the properties and expressions. Thus memorizing the properties is of utmost importance.