Does cosine rule apply to all triangles?

The Cosine Rule: Your Triangle’s Best Friend (No Matter What Shape It Is!)

Ever wrestled with a triangle that wasn’t a nice, neat right angle? We’ve all been there. That’s where the cosine rule comes to the rescue. Forget those limited right-triangle rules; the cosine rule is the Swiss Army knife of triangle solutions, working for any triangle you throw at it. So, yeah, it absolutely applies to all triangles. Period.

But what is this magical rule? Simply put, it connects the sides and angles of any triangle, no matter how wonky. Think of it as a souped-up version of the Pythagorean theorem, one that plays nice with all angles, not just 90-degree ones.

Here’s the gist: Imagine a triangle called ABC. Sides are a, b, c (lowercase), and the angles opposite them are A, B, C (uppercase). The cosine rule basically says you can find any side if you know the other two sides and the angle squished between them. The formulas look like this:

• a2 = b2 + c2 – 2bc * cos(A)
• b2 = a2 + c2 – 2ac * cos(B)
• c2 = a2 + b2 – 2ab * cos(C)

See? Not so scary. Basically, you plug in what you know, and out pops what you need to know.

Now, why does this work for every triangle? Good question! Unlike those basic trig functions (sine, cosine, tangent) that are all about right triangles, the cosine rule uses the cosine of an angle, which exists for any angle, big or small. It’s not picky!

And get this: the Pythagorean theorem? It’s actually just a special case of the cosine rule. Seriously! Remember how cos(90°) = 0? Well, if you stick a 90-degree angle into the cosine rule, the whole thing simplifies down to a2 + b2 = c2. Mind. Blown.

Okay, so it works. But where’s it useful? Everywhere! I’ve personally used it for everything from garden design (calculating fence lengths) to helping my friend figure out the best angle for his satellite dish. Here are a few more examples:

• Triangle Solver: Got two sides and the angle between them? Or all three sides? Cosine rule to the rescue!
• Navigation: Sailors and pilots use it to figure out distances and locations. Think triangulation!
• Engineering: Bridge builders use it to make sure those trusses are rock solid.
• Surveying: Mapping out land? The cosine rule is your friend.
• Physics: Even figuring out how a ball flies through the air can involve the cosine rule.

And hey, what if you need to find an angle instead of a side? No problem! Just rearrange the formulas:

• cos(A) = (b2 + c2 – a2) / 2bc
• cos(B) = (a2 + c2 – b2) / 2ac
• cos(C) = (a2 + b2 – c2) / 2ab

Then, just hit that “arccos” or “cos-1” button on your calculator, and boom, you’ve got your angle.

Let’s do a quick example to show you how easy it is. Imagine a triangle with sides of 8 and 11, with a 37-degree angle between them. Let’s find that missing side:

c2 = 82 + 112 – 2 * 8 * 11 * cos(37°)


c2 = 64 + 121 – 176 * 0.7986


c2 = 185 – 140.55


c2 = 44.45


c = √44.45 ≈ 6.67

So, the missing side is about 6.67 units long. Easy peasy, right?

Bottom line? The cosine rule is your go-to tool for tackling any triangle problem. It’s versatile, reliable, and honestly, kind of elegant in its simplicity. Master it, and you’ll be solving triangles like a pro in no time!