Question

How do you find the exact value of \[\sin 105{}^\circ \]?

Answer 1

Hint: In this question as it is given that the angle is \[105{}^\circ \] and we do not know the value of \[\sin 105{}^\circ \] then we can split the angle in two parts whose value are known to us and that are \[45{}^\circ \] and \[60{}^\circ \] . After this we can use the trigonometric formula for\[\sin \]function.

Formula used:
 \[\sin (A+B)=\sin A\cos B+\cos A\sin B\]

Complete step-by-step answer:
Here, comparing the above formula with the given question
\[\Rightarrow A+B=105{}^\circ { , A=60}{}^\circ { , B=45}{}^\circ \]
Now substituting these values in the formula
\[\sin (A+B)=\sin A\cos B+\cos A\sin B\]
\[\Rightarrow \sin (60{}^\circ +45{}^\circ )=\sin 60{}^\circ \cos 45{}^\circ +\cos 60{}^\circ \sin 45{}^\circ \]
And we know that
\[\Rightarrow \sin 60{}^\circ =\dfrac{\sqrt{3}}{2}{ , cos60}{}^\circ {=}\dfrac{1}{2}{ , sin45}{}^\circ {=cos45}{}^\circ {=}\dfrac{1}{\sqrt{2}}\]
Putting these values
\[\Rightarrow \dfrac{\sqrt{3}}{2}.\dfrac{1}{\sqrt{2}}+\dfrac{1}{2}.\dfrac{1}{\sqrt{2}}\]
\[\Rightarrow \dfrac{\sqrt{3}+1}{2\sqrt{2}}\]
Now rationalising the denominator
\[\Rightarrow \dfrac{\sqrt{2}+\sqrt{6}}{4}\]
Hence, \[\sin 105{}^\circ =\dfrac{\sqrt{2}+\sqrt{6}}{4}\]

Note: When we have to find the value of the trigonometric functions and the values of that angles are not known to us then we split the angles in such a way that the separate value of sin angles are known to us for example \[30{}^\circ ,45{}^\circ ,60{}^\circ ,90{}^\circ \].