What is the difference between tan and inverse tan?

Tan vs. Inverse Tan: Untangling These Trig Functions

Trigonometry. Just the word can send shivers down some spines, right? But honestly, once you get past the initial jargon, it’s a pretty fascinating world. And at the heart of it all are functions like tangent (tan) and its quirky cousin, inverse tangent (arctan, atan, or tan⁻¹). They’re related, sure, but they do totally different things. Let’s break it down in plain English, shall we?

Tangent: The Ratio Rockstar

First up, tangent. Think of a right-angled triangle – you know, the ones with that perfect 90-degree corner. The tangent of one of the other angles (let’s call it θ) is simply the ratio of the opposite side to the adjacent side. Picture it like this:

tan(θ) = Opposite / Adjacent

Easy peasy, right? You can also think of it as the sine divided by the cosine:

tan(θ) = sin(θ) / cos(θ)

Now, here’s where it gets a little tricky. You can plug pretty much any angle into the tangent function, but there are a few spots where it throws a fit – specifically, angles like 90 degrees, 270 degrees, and so on (odd multiples of π/2, if you want to get technical). At those points, the tangent is undefined because the cosine is zero, and you can’t divide by zero. The output, though, can be any real number, from way down in the negatives to way up in the positives.

Oh, and one more thing: tangent is a repeating function. It goes through the same cycle every π radians (or 180 degrees). So, tan(x + π) is always the same as tan(x). It’s like a trigonometric groundhog day!

Inverse Tangent: The Angle Finder

Okay, now for the inverse tangent, also known as arctan, atan, or tan⁻¹. This is where things get interesting. Forget calculating a ratio; instead, you start with the ratio and want to find the angle. In other words, it answers the question: “Hey, what angle has a tangent equal to this number?”. So, if tan(θ) = y, then arctan(y) = θ.

Think of it like this: tangent is like saying, “I have this angle, tell me the ratio.” Inverse tangent is like saying, “I have this ratio, tell me the angle.”

The inverse tangent takes any real number as input, but to keep things nice and tidy, we limit the output to angles between -π/2 and π/2 (-90 degrees and 90 degrees). This is the “principal value,” and it makes sure you always get a consistent answer.

And remember! tan⁻¹(x) is not the same as 1/tan(x). That’s the cotangent. The little “-1” is just a special symbol to tell you it’s the inverse function.

Tan vs. Arctan: The Key Differences

Let’s nail down the key differences:

FeatureTangent (tan)Inverse Tangent (arctan, atan, tan⁻¹)What it doesFinds the ratio of sides for a given angleFinds the angle for a given ratioInputAngleReal numberOutputReal numberAngle (in radians)DomainAlmost all angles (except odd multiples of π/2)Any real numberRangeAny real numberBetween -π/2 and π/2Repeating?Yes, every π radiansNope