Understanding When the Tangent Function Becomes Undefined

At which values of x is the tangent function undefined?

• A. ±90 and ±270
• B. 0 and 90
• C. ±180
• D. ±360

Answer: (A)

The tangent function becomes undefined at specific points where the cosine function, which is in the denominator of the tangent formula (tan(x) = sin(x)/cos(x)), equals zero. This occurs whenever x is an odd multiple of 90 degrees. More specifically, at x = 90 degrees (π/2 radians) and x = 270 degrees (3π/2 radians), the cosine function equals zero, resulting in the tangent function being undefined. This pattern continues with additional angles: at x = -90 degrees (-π/2 radians) and x = -270 degrees (-3π/2 radians), cosine is again zero, reinforcing that tangents are undefined at these points. Thus, the correct values of x that result in an undefined tangent function are indeed ±90 degrees and ±270 degrees, identifying where the cosine function equals zero. This illustrates the nature of the tangent function in trigonometry and its relationship with sine and cosine.

Ever found yourself asking, "Wait, when is the tangent function undefined?" Well, you're in the right place! Let's unravel the mystery surrounding the tangent function with some straightforward language and engaging explanations!

The Lowdown on Tangents

First off, if you’re brushing up for your GCSE Maths exam, this is a key concept! The tangent function, denoted as ( \tan(x) ), is all about ratios: it relates sine and cosine in a neat little formula: ( \tan(x) = \frac{\sin(x)}{\cos(x)} ). Pretty simple, right? However, this leads to an interesting quagmire when the cosine function equals zero because you simply can’t divide by zero!

So where exactly does that happen? Getting comfy? Let’s have a chat about those points where cosine hits a wall!

Finding the Trouble Spots: Odd Multiples of 90

The tangent function becomes undefined at specific points where the cosine function (the denominator in our equation) equals zero. Typically, this occurs at odd multiples of 90 degrees: ( +90, -90, +270, -270 ). Remember, it might seem a bit odd right now, but stay with me!

• At ( x = 90 ) degrees (or ( \frac{\pi}{2} ) radians), cosine equals zero, hence tangent goes poof!

• Similar magic takes place at ( x = 270 ) degrees (or ( \frac{3\pi}{2} ) radians).

• But wait, there's more – the same rings true for negative values! At ( -90 ) degrees (or ( -\frac{\pi}{2} ) radians) and ( -270 ) degrees (or ( -\frac{3\pi}{2} ) radians), guess what? Cosine equals zero again!

It’s like watching a recurring theme in your favorite series—those pivotal moments keep repeating, right?

Why Does This Matter in Real Life?

You might be thinking, “Okay, why should I care?” Well, understanding these intervals isn’t just about passing your GCSEs; it’s about laying the groundwork for more complex math down the road! Whether it’s in physics for waves or in engineering calculations, grasping these concepts makes you feel like you’re studying the secret language of the universe!

Wrapping It Up

So there you have it—the tangent function is undefined at ( ±90 ) degrees and ( ±270 ) degrees, where the cosine function takes a nosedive to zero. By keeping track of these angles, you not only up your game for your upcoming GCSE exam, but you also build a solid foundation for understanding a plethora of mathematical concepts later on. Now, go ahead and impress someone with your newfound knowledge—math can be cool after all!

Final Thoughts

Here's the thing: math isn’t just a bunch of formulas. It’s a fascinating web of connections and relationships. Keep exploring, questioning, and connecting those dots. Who knows? Next time you’re puzzling over a trigonometric function, you might just catch yourself smiling because you can see the bigger picture! And that’s what studying is all about. Happy calculating!