Is Cotangent an odd or even function?

Cotangent: Odd or Even? Let’s Clear Up the Confusion

So, you’re wondering whether cotangent is an odd or even function, huh? It’s a fair question! In the world of trigonometry, we like to categorize functions based on their symmetry. Knowing whether a function is even or odd can seriously simplify your calculations and give you a better feel for how it behaves. Let’s dive in and get this sorted out, once and for all.

Even vs. Odd: The Basic Rundown

First things first, let’s make sure we’re all on the same page about what “even” and “odd” actually mean when we’re talking about functions.

• Even Function: Think of an even function as a mirror image across the y-axis. Mathematically, that means f(-x) = f(x). Cosine? Classic example of an even trig function.
• Odd Function: Odd functions are a little more… twisty. They’re symmetrical around the origin. The rule? f(-x) = -f(x). Sine, tangent, and yes, our friend cotangent, all fall into this category.

Cotangent: The Cliff Notes Version

Okay, cotangent. What is it, exactly? Well, there are a few ways to think about it:

• It’s the flip-side of tangent: cot(x) = 1/tan(x).
• It’s cosine divided by sine: cot(x) = cos(x)/sin(x).
• Remember right triangles? It’s the adjacent side divided by the opposite side.

Cotangent’s a bit of a repeating character, with a period of π. It also has some interesting quirks, like vertical asymptotes popping up all over the place at multiples of π. Basically, it’s defined for every number except those multiples of pi. And its values? They can be any real number.

So, Is Cotangent Odd or Even? The Verdict

Alright, drumroll please… Cotangent is an odd function. End of story.

But hey, let’s prove it, just for kicks. Remember the definition? We need to show that cot(-x) = -cot(x). Here’s how it breaks down:

cot(-x) = cos(-x) / sin(-x)

Now, cosine’s even, so cos(-x) = cos(x). Sine’s odd, so sin(-x) = -sin(x). Plug that in, and you get:

cot(-x) = cos(x) / -sin(x) = – cos(x) / sin(x) = -cot(x)

Boom! There you have it. cot(-x) = -cot(x). Case closed.

A Picture’s Worth a Thousand Words

If you were to graph y = cot(x), you’d see that telltale symmetry around the origin. Spin the graph 180 degrees, and it looks exactly the same. That’s your visual confirmation that it’s an odd function. No y-axis symmetry here!

The Bottom Line

Cotangent is odd, through and through. Knowing this makes working with trig functions a whole lot easier. It’s one of those little facts that, once you get it, just sticks with you.