If \( y = \arctan(x) \), express \( x \) in terms of \( y \).

To express \( x \) in terms of \( y \) when \( y = \arctan(x) \), you can use the inverse tangent relationship. Specifically, since \( y = \arctan(x) \), it follows that \( x = \tan(y) \). This means you can find \( x \) by simply taking the tangent of the angle \( y \). Did you know that the arctangent function is particularly useful in real-world applications like navigation and physics? When calculating angles in a right triangle, for example, knowing one side length and the opposite angle allows you to find missing dimensions of various objects, making it essential in various engineering fields. Also, a common mistake when dealing with the arctan function is forgetting the range of the output. The value of \( \arctan(x) \) is restricted to the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). This means that when you convert back from \( y \) to \( x \), the tangent function can have multiple values, especially for angles beyond that range, leading to confusion if you’re not careful!