91Ó°ÊÓ

Inverse functions reverse the effect of the original function. If you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying the inverse function after the original function will get you back to the starting value. This means that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
In the context of trigonometry, inverse trigonometric functions help in finding angles when the value of the trigonometric ratio is known. For example, \( \tan^{-1}(x) \) tells us the angle whose tangent is \( x \).
In the given problem, \( \tan^{-1}(x) \) is used to isolate the variable \( x \). By applying the tangent function to both sides, the inverse tangent cancels out, leaving us with the value of \( x \).

Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. Solving such equations often requires finding the angles or other values that satisfy the equation.
In our example, the equation \( 3 \tan^{-1}(x) = \pi \) is a trigonometric equation. The solution starts by isolating the inverse tangent function by performing algebraic operations. By dividing both sides of the equation by 3, we simplify it to \( \tan^{-1}(x) = \frac{\pi}{3} \).
The next step involves eliminating the inverse tangent by applying the tangent function to both sides. This converts the equation into a simpler form where basic trigonometric values can be used.

The unit circle is a powerful tool in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate system. The unit circle helps in visualizing angles and trigonometric values.
Each point on the unit circle corresponds to an angle, and the coordinates of the point give the cosine (x-coordinate) and sine (y-coordinate) of that angle.
In the context of our problem, we used the unit circle to find that \( \tan\left( \frac{\pi}{3} \right) = \sqrt{3} \). On the unit circle, the angle \( \frac{\pi}{3} \) (or 60 degrees) corresponds to the point with coordinates \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \). The tangent of this angle is the ratio of the y-coordinate to the x-coordinate, which is \( \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \). Thus, we find that \( x = \sqrt{3} \).