Question

If in a triangle $\Delta ABC$, $a = 15$ , $b = 36$ , $c = 39$ then $\sin \dfrac{C}{2}$ is equal to
\[\left( 1 \right)\]$\dfrac{{\sqrt 3 }}{2}$
\[\left( 2 \right)\]$\dfrac{1}{2}$
\[\left( 3 \right)\]$\dfrac{1}{{\sqrt 2 }}$
\[\left( 4 \right)\]$\dfrac{{ - 1}}{{\sqrt 2 }}$

Answer 1

Hint: We have to find the value of $\sin \dfrac{C}{2}$ . We solve this question using the concept of cosine law of trigonometry . We should have the knowledge of the values of various trigonometric functions for different values of angles . We will find the value of angle of the triangle by putting the values in the formula of cosine law . And thereafter on simplifying the expressions we will get the value of $\sin \dfrac{C}{2}$ .

Complete step-by-step solution:
Given :
$a = 15$ , $b = 36$ and \[c{\text{ }} = {\text{ }}39\]
As we know that the formula of cosine law is given as :
$\cos C = \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}}$
Now , after putting the values we will get the value for angle $C$
Putting the values , we get the value as :
\[\cos C = \dfrac{{{{\left( {15} \right)}^2} + {{\left( {36} \right)}^2} - {{\left( {39} \right)}^2}}}{{2 \times 15 \times 36}}\]
After solving we get ,
\[\cos C = \dfrac{{225 + 1296 - 1521}}{{2 \times 15 \times 36}}\]
Further , we get the value as :
\[\cos C = 0\]
Using the values of cos function
We know that \[\cos \dfrac{\pi }{2} = 0\]
So , we can say that
\[\cos C = \cos \dfrac{\pi }{2}\]
Taking \[{\cos ^{ - 1}}\] both sides , we get
As , we know that \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\]
Now using the above formula , we get the value as :
\[C = \dfrac{\pi }{2}\]
Now , we have to find the value of \[\sin \dfrac{C}{2}\]
As , we have calculated that \[C = \dfrac{\pi }{2}\]
so,
\[\dfrac{C}{2} = \dfrac{\pi }{4}\]
Now , we get the value of \[\sin \dfrac{C}{2} = \sin \dfrac{\pi }{4}\]
Using the values of \[sin\]function
We know that \[\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\]
So ,
\[\sin \dfrac{C}{2} = \dfrac{1}{{\sqrt 2 }}\]
Thus the value of \[\sin \dfrac{C}{2} = \dfrac{1}{{\sqrt 2 }}\]
Hence , the correct option is \[\left( 3 \right)\] .

Note: While solving these types of questions we should take care of the range of the trigonometric functions, also we should take care about the value of the quadrant in which the angle would lie, as each trigonometric function has a different value in each quadrant.
The formula of triangle law of sine is given as :
\[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
The formula of triangle law of cosine is given as :
$\cos A = \dfrac{{{b^2} + {c^2} - {a^2}}}{{2bc}}$
Where \[A{\text{ }},{\text{ }}B{\text{ }},{\text{ }}C\]are the angles of the triangle\[ABC\]and \[a{\text{ }},{\text{ }}b{\text{ }},{\text{ }}c\]are the sides of the triangle\[ABC\]