Classic math proofs of the law of sine and the law of cosine

In this math tutorial, we explore the proofs behind the Law of Sine and the Law of Cosine, two fundamental trigonometric concepts. We provide clear explanations and visuals to help you better understand these concepts. By the end of the video, you'll have a solid foundation for further exploration in this fascinating field.

The formulas: 0:00
Proof of Law of Sine: 0:13
Proof of Law of Cosine: 4:20

[blackpenredpen]

The most power step in geometry/trigonometry: drop the perpendicular line!

[wayneliang]

Last year when our physics teacher taught us the definition of sine and cosine (he did it before we learn it in math and he only taught the definitions), the first test we get is to prove these two laws

[staysharp]

I love the way you teach things! I accept the rules when I am taught them but proving them makes it soooo much better for learning maths. Thank you so much BPRP

[dackid]

If you thought dropping the bass was cool, just you wait until you drop that perpendicular line.

[angelmendez-rivera]

You wanted us to prove the British flag theorem after the video. Here is my attempt.

Let the vertical line passing through the red point have a lower segment with length h1 and an upper segment with length h2, such that h1 + h2 = h, and let the horizontal line passing through that same point have a left segment with length w1 and a right segment with w2, such that w1 + w2 = w.

(w1)^2 + (h1)^2 = d^2
(w1)^2 + (h2)^2 = a^2
(w2)^2 + (h1)^2 = c^2
(w2)^2 + (h^2)^2 = b^2

a^2 + c^2 = (w1)^2 + (h2)^2 + (w2)^2 + (h1)^2 = (w1)^2 + (h1)^2 + (w2)^2 + (h2)^2 = d^2 + b^2 = b^2 + d^2.

Therefore, a^2 + c^2 = b^2 + d^2. Q.E.D.

[ledouble]

Be careful, the height can be outside of the triangle.
Si you have 2 cases to study.

For the 2nd theorem, you can use the scalar product by writing BC^2 = (BA + AC)^2
a^2 = c^2 + b^2 + 2BA.AC
a^2=c^2 + b^2 + 2c.b.cos(BA, AC)
a^2=c^2 + b^2 + 2c.b.cos(pi - a)
Then
a^2=c^2 + b^2 - 2c.b.cos(a)

Quite easy isn't it?

Hello from France, keep going!

[tubbyduckie]

Very helpful - I have a math exam tomorrow! Thank you so much!

[OonHan]

Ah yes, PERFECT timing. Currently learning trigo in school, and I am pretty confident this video will be a great resource for my class!

[rendanitshisikule]

Absolutely love the enthusiasm 👏🏾thank you for the help sir

[The-Cosmos]

I really loved how you gave that extra note that proving Law Of Cosines from pythagorean theorem is not circular reasoning because phythagorean already has like some 1000 proofs.
Say a set of theorems, X, is used to prove Pythogorean. So X=> Pythogorean => Law of cosine => Pythogorean

[drpkmath]

Laws of sine/cosine are frequently used in calculus if you take higher course than pre-cal courses so it would be very useful to have some understanding about the proof of them as bluepenredpen illustrates in the video right here. Great representation! Basic geometrical understanding is also a plus to understand this proof!

[ABCDEF-qxiw]

hey man, im from brazil. im watching your calculus's playlist and enjoying it. thanks for your videos, they really help me comprehend all the concepts

[anhdungtran]

Gonna watch all of your videos in this quarantine

[Chaosdude]

Your Law of Cosines proof is much, much more simple than the Stewart textbook, thank you!

Edit: I am struggling to do this for the other sides, but will get some rest and try again.

[erynn]

I love how mathematics although theoretically universal still has dialects. My teachers would have had a fit of I'd use "sin A", because A is a Point, not an Angle. The angle had to be labeled independently, mostly α for the angle near A, ⁠⁠β near B etc.

[kamaalhasan]

I came here to learn the derivations of these two laws, because I was lazy when I was in class and i missed what my teacher had said about this. Thanks teacher!

[asiyaali]

This video is awesome. It helped me so much. Thank you. Now I understand it better☺🙏

[sbmathsyt]

love your vidoes! You inspired me to make my own maths content, keep up the good work

[nazombie]

Great video my dude!!! I love watching videos where people can perfectly explain the pythagorean thereom and I learned a lot about law of sine/cosine

[Mihau_desu]

Really cool video. I originally proved law of sine the same way you presented, but in fact our text book provides method that also shows sinA/a = 1/2R where R is radius of a circle with points ABC on its perimeter(I don't know the word for it, English is not my first language).
Btw. I enjoyed the bonus exercise.
Keep up the good work. Cheers.