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 :
$si{n^{( - 1)}}( - x) = - si{n^{( - 1)}}(x),x \in [ - 1,1]$
$co{s^{( - 1)}}( - x) = \pi - co{s^{( - 1)}}(x),x \in [ - 1,1]$
$ta{n^{( - 1)}}( - x) = - ta{n^{( - 1)}}(x),x \in R$
$cose{c^{( - 1)}}( - x) = - cose{c^{( - 1)}}(x),|x| \geqslant 1$