Answer:
[tex]y=-\frac{2}{\sqrt{4-\pi^2}}x+Undefined[/tex]
Explanation:
Given the function:
[tex]f(x)=\cos^{-1}(x)[/tex]
First, determine the tangent point by evaluating f(x) at x=π/2:
[tex]f(\frac{\pi}{2})=\cos^{-1}(\frac{\pi}{2})=Undefined[/tex]
Note: f(π/2) is undefined because the domain of arccosine is (-1,1).
The tangent point is:
[tex](\frac{\pi}{2},Undefined)[/tex]
Next, find the slope of the tangent line. Begin by finding the derivative of f(x):
By the rules of derivative:
[tex]f^{\prime}(x)=-\frac{1}{\sqrt{1-x^2}}[/tex]
At x=π/2, find the value of the derivative:
[tex]\begin{gathered} f^{\prime}(\frac{\pi}{2})=-\frac{1}{\sqrt{1-(\frac{\pi}{2})^2}} \\ =-\frac{1}{\sqrt{1-\frac{\pi^2}{4}}} \\ =-\frac{1}{\sqrt{\frac{4-\pi^2}{4}}} \\ Slope,m=-\frac{2}{\sqrt{4-\pi^2}} \end{gathered}[/tex]
Thus, the equation of the tangent line is:
[tex]y=-\frac{2}{\sqrt{4-\pi^2}}x+Undefined[/tex]
