Tan-1 Calculator

The inverse tangent function, written mathematically as \( y = \arctan(x) \) or \( y = \tan^{-1}(x) \), reverses the standard tangent operation. If the tangent of an angle \( y \) gives a ratio \( x \), then the inverse tangent takes that ratio \( x \) and returns the original angle \( y \). In standard trigonometric convention, the output range of the arctangent function is restricted to the interval between \(-\pi/2\) and \(\pi/2\) radians (or \(-90^{\circ}\) and \(90^{\circ}\)) to ensure the function remains single-valued and invertible.

Let us walk through finding the inverse tangent when \( x = 1.0 \). First, recall that the tangent of an angle represents the ratio of the opposite side to the adjacent side in a right triangle. Setting up our equation gives \( y = \arctan(1.0) \). We ask ourselves: at what acute angle is the sine equal to the cosine, yielding a ratio of 1? That angle is \( 45^{\circ} \) or \(\pi/4\) radians. Therefore, the calculator output is \( 45^{\circ} \), successfully determining the angle corresponding to a slope of 1.

Always verify whether your calculator or application is set to output results in degrees or radians, as mixing these units is a common source of error. When working with two-dimensional coordinates where both \( x \) and \( y \) signs matter (such as in vector navigation), consider using the two-argument arctangent function (atan2) instead of standard tan-1 to correctly place the resulting angle in the proper quadrant.