How do you use cosine rule with 3 sides?

Cracking the Triangle Code: The Cosine Rule When You Know All the Sides

So, you’ve got a triangle. Not a right-angled one, mind you, but just a regular, run-of-the-mill triangle where you happen to know the length of all three sides. And you need to figure out the angles. Sounds tricky, right? Not with the cosine rule in your toolbox!

Think of the cosine rule, or law of cosines if you’re feeling fancy, as your secret weapon for tackling triangles. It’s a cornerstone of trigonometry, really, and while it works for any triangle, it’s especially handy when you’re staring at a triangle with three known sides (we call that SSS – Side, Side, Side) and scratching your head about the angles. Let’s break it down.

The Cosine Rule: A Quick Refresher

Basically, this rule links the sides of a triangle to the cosine of one of its angles. The classic formula looks like this:

c² = a² + b² – 2ab cos(C)

Where:

• a, b, and c are the lengths of the triangle’s sides.
• C is the angle sitting opposite side c.

But here’s the thing: when you already know all three sides and you’re hunting for an angle, it’s way easier to flip the formula around. We want to get that “cos(C)” all by itself. So, for angle C, we get:

cos(C) = (a² + b² – c²) / (2ab)

And naturally, the same idea applies if you’re after angles A or B:

cos(A) = (b² + c² – a²) / (2bc)

cos(B) = (a² + c² – b²) / (2ac)

See? Not so scary.

Step-by-Step: From Sides to Angles

Alright, let’s get practical. Here’s how you actually use this thing when you’re armed with three sides:

Let’s Do an Example

Okay, imagine a triangle where side a = 7, side b = 5, and side c = 10. Let’s find those angles, shall we?

A Few Things to Keep in Mind

• No Guesswork Here: Unlike some other triangle scenarios (SSA, I’m looking at you!), the cosine rule gives you a straight answer when you know all three sides. No ambiguous cases to worry about.
• Calculator Sanity: Double-check that your calculator is in degree mode or radian mode, depending on what you need. Trust me, it’s a common gotcha!
• Big Angle First: Here’s a pro tip I learned the hard way: when you’re given all three sides, try to find the largest angle first. Why? Because the cosine of angles bigger than 90° is negative. If you go after a smaller angle first and then try to use the Law of Sines, you might get tripped up trying to figure out if another angle is acute or obtuse. Save yourself the headache!
• Other Options? Sure, But…: Could you use Heron’s formula to find the area and then mess around with other trig formulas? Technically, yes. But honestly, the cosine rule is usually the most direct route.

Wrapping It Up

The cosine rule is a real workhorse when it comes to triangles. Once you get the hang of the formula and the steps, you’ll be able to confidently crack the code on any triangle where you know all three sides. Just remember to double-check your calculator settings, maybe aim for the biggest angle first, and you’ll be golden. Happy calculating!