Why Does Mathcad Treat atan and tan⁻¹ Differently?
Scenario / Question
I stumbled randomly on the following in PTC Mathcad Prime: it seems that Mathcad defines atan and tan⁻¹ differently. I thought that atan was the same as the inverse tangent, but that does not appear to be the case. Shouldn't atan and the inverse tangent be the same definition?
Overview
They are deliberately different. In PTC Mathcad Prime, atan(x) is arctan, the (limited) inverse function of tan, while tan(x)⁻¹ is the reciprocal 1/tan(x). This is correct and intended behavior: where a trigonometric function has a named inverse, the superscript ⁻¹ is read as an exponent. To get an angle back from a tangent value in Prime, always use atan.
Step-by-Step
Option 1: Get the inverse tangent (the angle)
- Use the atan function: Enter atan(x) where x is the tangent value. Prime returns the angle in radians, taken from the principal branch.
- Convert to degrees if required: Append the desired angle unit to the result, or format the result with the unit, so the value displays in degrees instead of radians.
- Do not substitute tan⁻¹ notation: Prime does not support tan⁻¹(x) as a name for the inverse function, as you cannot type that notation format in Mathcad.
Option 2: Get the reciprocal of the tangent
- Write the reciprocal explicitly: Enter 1/tan(x), (tan(x))⁻¹, or tan(x)⁻¹. All three return the reciprocal of the tangent.
- Prefer the fully bracketed form: (tan(x))⁻¹ is the clearest. The form tan(x)⁻¹ is perfectly legal but is often advised against, because it can be misread by other users as tan(x⁻¹).
Key Notes & Limitations
- atan(x) is the inverse function; tan(x)⁻¹ is the reciprocal 1/tan(x). Mixing them up is the single most common cause of an unexpected result here.
- The superscript-minus-one notation for an inverse trigonometric function was introduced by John Herschel in 1813, and it was often advised against because of exactly this confusion.
- Since 2009, the ISO 80000-2 standard specifies solely the "arc" prefix for these functions, so sin⁻¹ should never be used to mean arcsin. Prime adheres to that convention in meaning, though the names asin, acos, and atan are the Mathcad Prime representations of arcsin, arccos, and arctan.
- A useful sanity check: nobody reads exp⁻¹(x) or (eˣ)⁻¹ as the inverse of the exponential function, because that inverse has its own name, ln. Those forms mean e⁻ˣ or 1/exp(x). arctan versus tan⁻¹ works the same way.
- PTC Mathcad provides the full set of inverse trigonometric functions: asin(z), acos(z), atan(z), asec(z), acsc(z), and acot(z), all returning a value in radians.
Related Questions (FAQ)
Q: Is it a bug that tan⁻¹ does not return an angle in Mathcad Prime?
A: No. tan(x)⁻¹ is the reciprocal of the tangent, 1/tan(x), and that is the mathematically standard reading of the exponent. Use atan(x) when you want the angle.
Q: Which function do I use to get an angle from a tangent value?
A: Use atan(x). It returns the angle in radians from the principal branch, and you can apply a degree unit to the result to display it in degrees.
Q: Why does my calculator show tan⁻¹ on the arctan key?
A: That labelling follows the Herschel notation from 1813 rather than ISO 80000-2 from 2009. It is widespread on calculators and in some publications, but it conflicts with the exponent reading used by Mathcad Prime.
Q: Can I write the reciprocal as tan(x)⁻¹ safely?
A: Yes, it is legal in Prime, but (tan(x))⁻¹ or 1/tan(x) is clearer, because tan(x)⁻¹ can be misread as tan(x⁻¹).
Links & Resources
- Inverse Trigonometric Functions, PTC Mathcad Help Center (support.ptc.com)
- About Trigonometric, Log, and Exponential Functions, PTC Mathcad Help Center (support.ptc.com)
- Example: Inverse Trigonometric Functions, PTC Mathcad Help Center (support.ptc.com)
- Inverse of Tan Theta, related community thread (community.ptc.com) (7 replies, 38,888 views)
- atan vs tan^-1, original community thread (community.ptc.com) (4 replies, 53,187 views, solved)