Question

How do you use the half angle identity to find $\sin {105^ \circ }$ ?

Answer 1

Hint : We use half angle formulae for sine function to find the value of the given trigonometric function. The formula or identities are universally true and holds good for any angle.
Use the half angle formula
$\sin \left( {\dfrac{\theta }{2}} \right) = \pm \sqrt {\dfrac{{1 - \cos \theta }}{2}}$

Complete step by step solution:
In the case we want to find $\sin \left( {{{105}^ \circ }} \right)$ so that’s what we want $\sin \left( {\dfrac{\theta }{2}} \right)$ to equal .
To find what our $\theta$ is , set these to equal to each other.
$\sin \left( {{{105}^ \circ }} \right) = \sin \left( {\dfrac{\theta }{2}} \right)$
$\Rightarrow {105^ \circ } = \dfrac{\theta }{2}$
$\Rightarrow {210^ \circ } = \theta$
This is our $\theta$ . Now, we can use the half angle formula.
$\sin \left( {{{105}^ \circ }} \right)$
$= \sin \left( {{{\dfrac{{210}}{2}}^ \circ }} \right)$
$= \pm \sqrt {\dfrac{{1 - \cos \left( {{{210}^ \circ }} \right)}}{2}}$
$= \pm \sqrt {\dfrac{{1 + \dfrac{{\sqrt 3 }}{2}}}{2}}$
$= \pm \sqrt {\dfrac{{1 + \sqrt 3 }}{4}}$
$= \pm \dfrac{{\sqrt {2 + \sqrt 3 } }}{{\sqrt 4 }}$
$= \pm \dfrac{{\sqrt {2 + \sqrt 3 } }}{2}$ $$
Since ${105^ \circ }$ is in the second quadrant, we know that in the second quadrant $\sin \theta$ be positive.
Therefore,
$\sin {105^ \circ } = \pm \dfrac{{\sqrt {2 + \sqrt 3 } }}{2}$
So, the correct answer is “Option $\sin {105^ \circ } = \pm \dfrac{{\sqrt {2 + \sqrt 3 } }}{2}$ ”.

Note : We will take a negative answer when $\theta$ lies in ${3^{rd}}$ or ${4^{th}}$ quadrant. The abbreviation for 'all sin cos tan' rule in trigonometry is ASTC . It can be memorized as "All Students Take Calculus". The first letter of the first word in this phrase is 'A'. This may be taken to indicate that all trigonometric ratios in the first quadrant are positive .