# The coolest fact in trigonometry that you learn way too late

• Femme_physics
In summary, the conversation discusses the law of cosines and its connection to the Pythagorean theorem in trigonometry. It also mentions the interesting fact that if A=0, cos A=1 and the law of cosines becomes the Pythagorean theorem. The conversation ends with a link to a website with more interesting math gems.
Femme_physics
Gold Member
The coolest fact in trigonometry that you learn way too late...

So I just spent the last 12 hours learning trig (occasional food breaks).

I just want to share with you something really, really, really, really, awesome.

Check this out, you're going to be blown away

The law of cosines, applied to a right triangle is c^2 = a^2+b^2 - 2abCosA Whereas A is equal to zero!

So the law of cosines for a right triangle is, in fact, the Pythagorean theorm! Math is sooooooooo awesome. Why did it only take me the last quarter of the textbook to learn this fact?

Perhaps I'm missing something here, but if A = 0, cos A = 1, and your statement comes in as

$$c^2 = a^2 + b^2 - 2ab$$

which is not the Pythagorean Theorem.

I mean, A is 90 degrees and therefor 0 is equal to 0!

Sorry

Dory said:
Check this out, you're going to be blown away

The law of cosines, applied to a right triangle is c^2 = a^2+b^2 - 2abCosA Whereas A is equal to zero!

So the law of cosines for a right triangle is, in fact, the Pythagorean theorm! Math is sooooooooo awesome. Why did it only take me the last quarter of the textbook to learn this fact?

yep it's awesome

Perhaps I'm missing something here, but if A = 0, cos A = 1, and your statement comes in as

$$c^2 = a^2 + b^2 - 2ab$$

which is not the Pythagorean Theorem.

Of course, this value is interesting too, as it's the theorem that if you have three collinear points:
A --- B ---- C​
Then the length of the whole (b) is the sum of the lengths of the two parts (a and c).

Dory said:
I mean, A is 90 degrees and therefor 0 is equal to 0!

Sorry

Don't worry - I had a feeling this is what you meant - but I wanted to make sure.

You'll find lots of cool things in your math classes if you take the time to look around.

Dory said:
I mean, A is 90 degrees and therefor 0 is equal to 0!
You don't need the law of cosines to conclude 0 = 0.

Awesome link granpa! And thanks for the replies and humoring me :)

## What is the coolest fact in trigonometry that you learn way too late?

The coolest fact in trigonometry that you learn way too late is the law of cosines. It allows you to find the length of a side of a triangle by using the cosine function, even if you don't know any angles.

## How does the law of cosines work?

The law of cosines is based on the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The law of cosines extends this concept to non-right triangles by using the cosine function to calculate the length of a side.

## Why is the law of cosines important?

The law of cosines is important because it allows you to solve for sides and angles of a triangle that may not be right triangles. This is especially useful in real-world applications, such as surveying and navigation, where you may encounter non-right triangles.

## Can the law of cosines be used with any triangle?

Yes, the law of cosines can be used with any triangle, regardless of its type (acute, right, or obtuse). However, it is most commonly used with obtuse triangles, as this is where it differs from the Pythagorean theorem.

## Is the law of cosines the only way to find the length of a side of a triangle?

No, there are other trigonometric laws and formulas that can be used to find the length of a side of a triangle, such as the Pythagorean theorem, the law of sines, and the law of tangents. However, the law of cosines is the most versatile, as it can be used in any type of triangle.

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