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Question: Find the value of x for the given inverse trigonometric function \(4{\sin ^{ - 1}}x + {\cos ^{ - 1}}...

Find the value of x for the given inverse trigonometric function 4sin⁡−1x+cos⁡−1x=π4{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \pi .

Explanation

Solution

Hint: In order to solve the problem first try to convert these two trigonometric functions into one by the use of different inverse trigonometric identities and then move on with the simplification part and calculate the value of x.

Complete step-by-step answer:
Given equation is: 4sin⁡−1x+cos⁡−1x=π4{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \pi
We need to find x from this equation,
As we know that
sin⁡−1x+cos⁡−1x=π2 ⇒cos⁡−1x=π2−sin⁡−1x  {\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \dfrac{\pi }{2} \\\ \Rightarrow {\cos ^{ - 1}}x = \dfrac{\pi }{2} - {\sin ^{ - 1}}x \\\
Let us substitute the value from above formula into the given equation

⇒4sin⁡−1x+cos⁡−1x=π ⇒4sin⁡−1x+(π2−sin⁡−1x)=π  \Rightarrow 4{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \pi \\\ \Rightarrow 4{\sin ^{ - 1}}x + \left( {\dfrac{\pi }{2} - {{\sin }^{ - 1}}x} \right) = \pi \\\

Now let us simplify the equation to find the value of x

⇒3sin⁡−1x+π2=π ⇒3sin⁡−1x=π−π2 ⇒3sin⁡−1x=π2 ⇒sin⁡−1x=13×π2 ⇒sin⁡−1x=π6  \Rightarrow 3{\sin ^{ - 1}}x + \dfrac{\pi }{2} = \pi \\\ \Rightarrow 3{\sin ^{ - 1}}x = \pi - \dfrac{\pi }{2} \\\ \Rightarrow 3{\sin ^{ - 1}}x = \dfrac{\pi }{2} \\\ \Rightarrow {\sin ^{ - 1}}x = \dfrac{1}{3} \times \dfrac{\pi }{2} \\\ \Rightarrow {\sin ^{ - 1}}x = \dfrac{\pi }{6} \\\

Now, let us bring the sin⁡−1{\sin ^{ - 1}} function from the LHS to the RHS. So it will convert to sin⁡\sin function.
⇒x=sin⁡(π6) ⇒x=sin⁡(18006) [∵π=1800] ⇒x=sin⁡(300)  \Rightarrow x = \sin \left( {\dfrac{\pi }{6}} \right) \\\ \Rightarrow x = \sin \left( {\dfrac{{{{180}^0}}}{6}} \right){\text{ }}\left[ {\because \pi = {{180}^0}} \right] \\\ \Rightarrow x = \sin \left( {{{30}^0}} \right) \\\
As we know the value of sin⁡300\sin {30^0} from the trigonometric table. So we will directly substitute the value.
x=sin⁡300=12 ⇒x=12  x = \sin {30^0} = \dfrac{1}{2} \\\ \Rightarrow x = \dfrac{1}{2} \\\
Hence, the value of x is 12\dfrac{1}{2} .

Note: In order to solve such problems students must remember the formulas for the inverse trigonometry. Also students must learn the values of the trigonometric terms for some common angles. This question would have been impossible to solve if we would not have converted the two functions into one so students must recognize which functions to remove.