The Most Useful Trig Identity - Harmonic Addition Theorem

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Today we prove, that the sum of sine and cosine waves always results in a shifted sine or cosine, namely a*cos(phi)+b*sin(phi)=r*cos(phi+alpha) or a*cos(phi)+b*sin(phi)=r*sin(phi+alpha) :) Enjoy! =D

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Postfach 11 15
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[PapaFlammy]

*_gibe sub if u liekdd :vvv_* Here are all of today's relevant links btw

[lemqnade]

I'm in 11th grade, studying trigonometric equations and keep asking how a sin x + b cos x = k cos (x+α) came. and now papa flammy has explained it clearly. thanks alot

[mastershooter]

"Flammy's Wood"

Lmaooo that's the best name ever!

[nate]

There's also a cute shortcut:

Write a cos(x) +b sin(x) as (a, b). (cos x, sin x). Now this is |(a, b) | |(cos x, sin x) | cos (angle)

The first is Sqrt(a^2+b^2), the second ist 1.finally note that the angle between (a, b) and the x axis is arctan(b/a), the angle between (cos x, sin x) and the x axis is x so that the angle in between is x-arctan(b/a)

It's a bit hand wavy but quite lovely :)

[youngmathematician]

I derived it for the sine instead of the cosine and got eta=+-sqrt(a^2+b^2) and t=arctan(a/b). Therefore,

[CallOFDutyMVP]

Please more nonlinear dynamics! I'm talking Lorenz Equations and the 4-dimensional Competitive Lotka-Volterra equations (if you're feeling woody)

[hugodaniel]

We use this property for harmonic vibrations in mechanics.

[PiotrWieczorek]

I love how at 6:44 you've felt compelled to explain why -1 cannot be equal to 0. :D

[garvett]

Flammy mah’ boi I have stumbled upon a really fun problem in Courant’s textbook on Differential and Integral Calculus:

The numbers 𝛾₁ and 𝛾₂ are the direction cosines of a line; that is, 𝛾₁² + 𝛾₂² = 1. Similarly, 𝜂₁² + 𝜂₂² = 1. Prove that the equation 𝛾₁𝜂₁ + 𝛾₂𝜂₂ = 1 implies that 𝛾₁ = 𝜂₁ and 𝛾₂ = 𝜂₂ .

It is particularly juicy if you solve it using trigonometry. If that does not interest you, I should propose that you make a video on direction cosines of lines. Might be fun!

[hectorgalva]

Hahaha the introduction image... "2021 Olympic gold medals, by planet" made my day😂😂😂😂

[edwinjoy]

When I first sumbled upon this identity in my communication systems class, this was my intuition on how this should be true. I looked at sin and cos to be unit orthogonal vectors (because of course I was done with Fourier Series! 😉) represented as x and y axes respectively on the Cartesian plane. a and b are the scaling factors. Now imagine representing it in polar form. The magnitude of resulting vector is sqrt(a^2 + b^2) and the angle made by the resultant with the x axis (or cos) is atan(b/a). As you can see the magnitude part is portrayed in the identity, the angle is portrayed as a phase shift of atan(b/a) from the cos(pi). BOOM!

[robertcitrus]

My friend: What's your favorite large number
Me: 9.2537817256*10^(29)
My friend: Why?
Me: take the natural log of it...

[cyto]

Very satisfying video somehow, it feels so good when all variables get substituted with the basic ones :)

[zaydabbas]

I was literally thinking of top 10 trig identities last night and this was my #1

[shridharshendye]

How this guy flips from Fred Wesley to Professor Lupin when he starts teaching😂

[nikhilnagaria]

0:08 an event probabilistically equivalent to the observation of Halley's comet aka Papa not being high while doing the intro

[einsteingonzalez]

6:33
How sin(x) is pronounced:

Normal People: [saɪn]
Dr. Fehlau: [sən]

[sharpnova]

i learned this really recently to teach to a student of mine. pretty cool and a little counterintuitive to me

[kenny]

Love your videos man, just commenting to show support!

[ashveet]

I was just studying this and needed help. Thank you so much for the beautiful video. I hope your Italy vacation was amazing.😀😀😀