Question

How do you solve using the sum and difference formulae for \[\sin \left( {\dfrac{{11\pi }}{{12}}} \right)\] ?

Answer 1

Hint:Here the given term can be evaluated directly but the question need to solve by sum and difference method which include the breaking of angle into two so that you can solve easily, in sum and difference method we see to simplify the bigger angles into small as per our convenience and to achieve the result easily.

Complete step by step answer:
The given question is to solve for \[\sin \left( {\dfrac{{11\pi }}{{12}}} \right)\]. Here we are instructed to solve by sum and difference formulae, for this question we are going to use difference formulae such that we can easily get the result with that, for solving the angle we are going to break the angle into difference of two, which is twelve subtract one, from this pair of angles one fraction that is twelve will be resolved and we are left with pie only for that fraction, and then only one fraction angle will be left to solve, on solving we get:
\[\sin \left( {\dfrac{{11\pi }}{{12}}} \right) = \sin \left( {\dfrac{{12\pi - 1\pi }}{{12}}} \right) \\
\Rightarrow \sin \left( {\dfrac{{11\pi }}{{12}}} \right)= \sin \left( {\pi - \dfrac{\pi }{{12}}} \right) \\
\Rightarrow \sin \left( {\dfrac{{11\pi }}{{12}}} \right)= \sin \left( {\dfrac{\pi }{{12}}} \right)\left[ {\sin (\pi - \theta )= \sin \theta } \right] \\
\therefore \sin \left( {\dfrac{\pi }{{12}}} \right) = 0.6528 \\ \]
Here we got the final value for the given question.

Note:We have break the angle such that one fraction dissolve and the value of other left angle can be found easily, you can use here sum also for solving the angle but by using difference of angles it was simple to go with , rest the final answer will be obtained same in both case. Here for this question we can also plot a graph to get the answer.