Question

If $\sin ({{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x)=1$ , then find the value of x.

Answer 1

Hint: To solve this inverse trigonometric question, what we will do is, we will first find the value of ${{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x$in radian and then using the inverse trigonometric property which is ${{\sin }^{-1}}x+{{\cos }^{-1}}x=\dfrac{\pi }{2}$, we will find the value of x.

Complete step by step answer:
Now, before we start solving this question, let us see values of $\sin x$for different degree angles and we will also see what are some inverse trigonometric formulas.
Now, value of $\sin x$ at 0 is 0, value of $\sin x$ at $\dfrac{\pi }{6}$ is $\dfrac{1}{2}$, value of $\sin x$ at $\dfrac{\pi }{4}$ is $\dfrac{1}{\sqrt{2}}$, value of $\sin x$ at $\dfrac{\pi }{3}$ is $\dfrac{\sqrt{3}}{2}$ and value of $\sin x$ at $\dfrac{\pi }{2}$ is 1.
Now, some inverse trigonometric identities are,
${{\sin }^{-1}}x+{{\cos }^{-1}}x=\dfrac{\pi }{2}$, ${{\tan }^{-1}}x+{{\cot }^{-1}}x=\dfrac{\pi }{2}$ and ${{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x=\dfrac{\pi }{2}$.
Now, in question it is given that $\sin ({{\sin }^{-1}}x+{{\cos }^{-1}}x)=1$
So, we can re – write $\sin ({{\sin }^{-1}}x+{{\cos }^{-1}}x)=1$ as
$\sin ({{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x)=\sin \left( \dfrac{\pi }{2} \right)$, as we discussed above that value of $\sin x$ at $\dfrac{\pi }{2}$ is 1.
As on right side and left side we have function of sin, so we can compare the inputs,
So, on comparing, we get
${{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x=\dfrac{\pi }{2}$
Now, we know that ${{\sin }^{-1}}x+{{\cos }^{-1}}x=\dfrac{\pi }{2}$,
So, comparing ${{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x=\dfrac{\pi }{2}$ with ${{\sin }^{-1}}x+{{\cos }^{-1}}x=\dfrac{\pi }{2}$, we get
$x=\dfrac{1}{5}$
Hence, the value of $\sin ({{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x)$ is equal to one if x is equals to $\dfrac{1}{5}$ that is $\sin ({{\sin }^{-1}}\dfrac{1}{5}+{{\cos }^{-1}}x)=1$for $x=\dfrac{1}{5}$.

Note:
While solving questions based on inverse trigonometric function, we must know all the properties of inverse trigonometric function as well as trigonometric function because some questions are tricky, which can be solved easily with help of these identities. While solving the questions and evaluating values of x, try to avoid calculation error as this will make you stuck in between of the solution or may give you incorrect answers.