Question

Solve $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = $ $ $
\[\left( 1 \right)\] \[\dfrac{{71}}{{125}}\]
\[\left( 2 \right)\] \[\dfrac{{74}}{{125}}\]
\[\left( 3 \right)\] \[\dfrac{3}{5}\]
\[\left( 4 \right)\] \[\dfrac{1}{2}\]

Answer 1

Hint: We have to find the value of $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] $ . We solve this by using the concept of inverse trigonometry and using the sin triple angle formula . We would simply equate the value of $ si{n^{( - 1)}} $ to a variable and apply the sin triple angle formula This will give the required .

Complete step-by-step answer:
All the trigonometric functions are classified into two categories or types as either sine function or cosine function . All the functions which lie in the category of sine functions are sin , cosec and tan functions on the other hand the functions which lie in the category of cosine functions are cos , sec and cot functions . The trigonometric functions are classified into these two categories on the basis of their property which is stated as : when the value of angle is substituted by the negative value of the angle then we get the negative value for the functions in the sine family and a positive value for the functions in the cosine family .
Given : $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] $ ——(1)
Let us consider that
 $ si{n^{( - 1)}}(\dfrac{1}{5}) = x $
Taking sin both sides , we get
\[\dfrac{1}{5} = {\text{ }}sin{\text{ }}x\]——(2)
Now , putting value (2) in equation (1)
 $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = sin[3si{n^{( - 1)}}\sin x] $
Also , we know that $ [si{n^{( - 1)}}\sin x] = x $
So ,
 $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = sin(3x) $
 Now , using sin triple angle formula
 $ Sin3x = 3sinx - 4si{n^3}x $
We get ,
 $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = 3\sin x - 4si{n^3}x $ ——(3)
Substituting value of \[sin{\text{ }} = \dfrac{1}{5}\] in (3)
 $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = 3 \times (\dfrac{1}{5}) - 4{(\dfrac{1}{5})^3} $
 $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = (\dfrac{3}{5}) - (\dfrac{4}{{125}}) $
On solving we get,
 $ sin[3si{n^{( - 1)}}(\dfrac{1}{5})] = (\dfrac{{71}}{{25}}) $
Thus , the correct option is \[\left( 1 \right)\]
So, the correct answer is “Option 1”.

Note: We used the concept of inverse trigonometric functions and formula of sin triple angle .
Also various inverse functions are :
 $ si{n^{( - 1)}}( - x) = - si{n^{( - 1)}}(x),x \in [ - 1,1] $
 $ co{s^{( - 1)}}( - x) = \pi - co{s^{( - 1)}}(x),x \in [ - 1,1] $
  $ ta{n^{( - 1)}}( - x) = - ta{n^{( - 1)}}(x),x \in R $
 $ cose{c^{( - 1)}}( - x) = - cose{c^{( - 1)}}(x),|x| \geqslant 1 $