What is the formula of tan 90 Theta?

Cracking the Code: What’s the Deal with tan(90° ± θ)?

Trigonometry can feel like navigating a maze sometimes, right? But once you start to see the connections, it’s actually pretty cool. One of those key connections involves understanding how the tangent function behaves around 90 degrees. So, let’s break down the formulas for tan(90° – θ) and tan(90° + θ) – trust me, it’s simpler than it sounds!

Quick Refresher: Tangent Basics

Before we jump in, let’s quickly remind ourselves what the tangent function is all about. Imagine a right-angled triangle. The tangent of one of the acute angles (we’ll call it θ) is just the ratio of the opposite side to the adjacent side. Simple as that! You can also think of it as sin(θ) divided by cos(θ). This will be handy later.

Unveiling the Formula for tan(90° – θ)

Okay, here’s the first formula:

tan(90° – θ) = cot(θ)

“Cot,” short for cotangent, is simply the flip-side of the tangent function. Think of it as 1/tan(θ). So, what this formula is telling us is that the tangent of an angle’s complement (that is, 90 degrees minus the angle) is equal to the cotangent of the original angle. Makes sense, right? It’s all about those complementary angle relationships we learned back in geometry.

Example:


Let’s say we want to find tan(90°- 45°).


Since tan(90°- θ )= cot θ,


tan(90°- 45°) = cot 45° = 1. Easy peasy!

Decoding the Formula for tan(90° + θ)

Now, let’s tackle the second formula:

tan(90° + θ) = -cot(θ)

Spot the difference? That sneaky little negative sign! This is because when you go past 90 degrees into the second quadrant, the tangent function turns negative.

Why the negative sign?


Well, tan(90° + θ) = sin(90° + θ) / cos(90° + θ). And since sin(90° + θ) = cos(θ) and cos(90° + θ) = -sin(θ),


tan(90° + θ) = cos(θ) / -sin(θ) = -cot(θ). There you have it!

Example:


So, if you ever need to figure out tan (90° + θ), just remember it’s equal to – cot θ.

The Curious Case of tan 90°

Ever wondered what tan 90° is? Well, here’s a fun fact: it’s undefined! Remember that tangent is sin/cos. At 90 degrees, you’re dividing 1 by 0, and that’s a big no-no in the math world. As you inch closer and closer to 90° on the unit circle, the value of tangent just keeps growing and growing, heading towards infinity. Wild, huh?

Why Bother with These Formulas?

These formulas aren’t just abstract math; they’re super useful in simplifying tricky trigonometric expressions and solving equations. I’ve used them countless times in calculus problems, physics simulations, and even some engineering projects involving things that oscillate or repeat. Seriously, knowing these identities can turn a monster equation into something much more manageable.

Final Thoughts

So, there you have it – the lowdown on tan(90° – θ) and tan(90° + θ). Understanding these formulas is like adding a new tool to your trigonometry toolbox. Just remember tan(90° – θ) = cot(θ) and tan(90° + θ) = -cot(θ). And don’t forget that tan 90° is a mathematical no-go zone! Keep practicing, and you’ll be cracking those trig problems in no time.