Question

How do you find the exact value of$\sin \left( {a + b} \right)$?

Answer 1

Hint: Split the angle into two and apply $\sin \left( {a + b} \right)$ formula.
The very first step that we need to do in this type of problem is to divide the existing angle into two angles in such a way that the sin and cos values of those angles must be known. Therefore, here we will write 105 as 60+45. After this step, we will apply the $\sin \left( {a + b} \right)$formula which is given by $\sin a\cos b + \cos a\sin b$where $a = 60\,$and$b = 45$. After solving the formula with the values we will get the answer as $\dfrac{{1 + \sqrt 3 }}{{2\sqrt 2 }}$.

Complete step by step answer:
Here, the given value is $\sin \left( {a + b} \right)$. The first step that we need to do here is to divide the value $\dfrac{{7\pi }}{{12}}$ into two angles. Now, just to simplify,
$
\dfrac{{7\pi }}{{12}} = \dfrac{{7 \times 180}}{{12}} \\
\Rightarrow\dfrac{{7\pi }}{{12}}= {105^ \circ } \\
$
Therefore, $\sin \left( {a + b} \right)$ can also be written as $\sin (105)$
So, dividing the angle 105 into two angles. Let’s say 60 and 45.
$\sin (105) = \sin (60 + 45)$
Now, applying $\sin \left( {a + b} \right)$ formula into the above form we can further solve this problem as:-
$\sin (a + b) = \sin a\cos b + \cos a\sin b$
So,
$\sin (60 + 45) = \sin 60\cos 45 + \cos 45\sin 60$
The values of respective sin and cos angles are already known. Which are:-
\[
\sin (45) = \dfrac{1}{{\sqrt 2 }} \\
\Rightarrow\sin (60) = \dfrac{{\sqrt 3 }}{2} \\
\Rightarrow\cos (45) = \dfrac{1}{{\sqrt 2 }} \\
\Rightarrow\cos (60) = \dfrac{1}{2} \\
\]
Putting the values of sin and cos respectively in the $\sin \left( {a + b} \right)$ formula, we will get
\[
\sin (a + b) = \sin a\cos b + \cos a\sin b \\
\Rightarrow\sin (105)= \sin (45)\cos (60) + \cos (45)\sin (60) \\
\Rightarrow\sin (105)= \left[ {\dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{2}} \right] + \left[ {\dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2}} \right] \\
\Rightarrow\sin (105)= \left[ {\dfrac{1}{{2\sqrt 2 }}} \right] + \left[ {\dfrac{{\sqrt 3 }}{{2\sqrt 2 }}} \right] \\
\therefore\sin (105)= \dfrac{{1 + \sqrt 3 }}{{2\sqrt 2 }} \\\]
Therefore, the exact value of $\sin \left( {\dfrac{{7\pi }}{{12}}} \right)$is $\dfrac{{1 + \sqrt 3 }}{{2\sqrt 2 }}$.

Note:The formulas of $\sin (a + b)$and other trigonometric functions must be remembered by heart as these formulas will help you to solve problems like this. Also, make sure, whenever you divide the angle, divide it in such a way that the resulting two angle’s sin/cos/tan etc exact values are known. Which means, generally the addition or subtraction of those angles will include either 30/45/60/90 degrees.