Question

Find the value of the trigonometric expression : \[\sin \dfrac{\pi }{8}\].
A. \[\sqrt {\dfrac{{\sqrt 2 - 1}}{{2\sqrt 2 }}} \]
B. Does not exist
C. Cannot be determined
D. None of these

Answer 1

Hint: The given problem revolves around the concepts of finding the respective value of the given equation/expression by the brief solutions. As a result, to find such value first of all considering the formula of trigonometric ratio for half angle that is of \[\cos \dfrac{\theta }{2} = 1 - 2{\sin ^2}\dfrac{\theta }{2}\] respectively. Hence, then substituting or considering the value of angle in the given expression present i.e. \[\dfrac{\pi }{8}\], the desired value is obtained.

Complete step by step answer:
Since, we have given that
\[\sin \dfrac{\pi }{8}\]
As a result, we have asked to solve this expression to find the respective value
Hence,
We know that,
Trigonometric ratios for half angles that is, \[\cos \dfrac{\theta }{2} = 1 - 2{\sin ^2}\dfrac{\theta }{2}\] … (i)
Hence, relating the given expression so as to get the desire value considering \[\theta = \dfrac{\pi }{8}\], we get
Since, we have considered ‘\[\dfrac{\theta }{2} = \dfrac{\pi }{8}\]’ respectively
But, for ‘\[\cos \theta \]’ (to get the exact value from the trigonometric table)
Hence, calculating the value ’\[\theta \]’, we get
\[\dfrac{\theta }{2} = \dfrac{\pi }{8}\]
\[\theta = \dfrac{\pi }{8} \times 2 = \dfrac{\pi }{4}\]
Hence, the equation (i) becomes
\[ \Rightarrow \cos \dfrac{\pi }{4} = 1 - 2{\sin ^2}\dfrac{\pi }{8}\]
Now, we know that \[\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\]
Hence, substituting in the above equation, we get
\[ \Rightarrow \dfrac{1}{{\sqrt 2 }} = 1 - 2{\sin ^2}\dfrac{\pi }{8}\]
Solving the equation mathematically that is taking the constants on one side so as to get the required value of \[\sin \dfrac{\pi }{8}\], we get
\[ \Rightarrow 2{\sin ^2}\dfrac{\pi }{8} = 1 - \dfrac{1}{{\sqrt 2 }}\]
\[ \Rightarrow 2{\sin ^2}\dfrac{\pi }{8} = \dfrac{{\sqrt 2 - 1}}{{\sqrt 2 }}\]
Now, hence dividing the equation by \[2\], we get
\[ \Rightarrow {\sin ^2}\dfrac{\pi }{8} = \dfrac{{\sqrt 2 - 1}}{{2\sqrt 2 }}\]
As a result,
Taking the square roots, we get
\[ \Rightarrow \sin \dfrac{\pi }{8} = \sqrt {\dfrac{{\sqrt 2 - 1}}{{2\sqrt 2 }}} \]
\[\therefore \] Option (A) is correct.

Note:
One must able to relate the half angle formulae with the standard angle (i.e. \[{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ },{90^ \circ }\], etc.) so as to put the direct value in the respective solution likewise here we have considered \[\cos {45^ \circ } = \cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\] as the half angle considered was \[\dfrac{\pi }{8}\] respectively. As a result of these complications, remember all the trigonometric formulae, especially double angle, half angle, triple angle, factorization and defactorization formulae, etc., so as to be sure of our final answer.