Derivation of Sum and Difference Identities

The derivation of the sum and difference identities for cosine and sine.

[the-dan-signal]

My professor didn't have time to derive these equations for the class and we had to settle for memorizing them. It feels so good to know where they come from, now!

[AndrewSFTSN]

That description of cosine as an "even" function and sine as an "odd" function really cleared up some other things for me, let alone these identities. Thanks a lot.

[pamperedpanda]

This video is truly a timeless classic. It was puzzling to me why the sum/diff formula is the way it is, and so I was doing it mindlessly. Solving trig problems make more sense now. Thanks!

[Reivivus]

This was beautiful! You actually know what you are talking about. 

[alexandrianova]

Excellent, a rare resource that is truly needed.

[Flamingponycat]

really helpful, helped me understand the equations I've been learning in school.

[susie.rosalie]

You are my hero. Like- I'm supposed to do a report on this topic and I was so lost until I found this vid. Thank you. Thank you soo much. Like seriously, you saved me.

[helixgaming]

I really appreciate the video man. I understand so much more about the subtraction/addition identities. Thanks!

[justpassingby]

This video was so much helpful! I really can't thank you enough, sir. By the way sir, why have you stopped making videos?Your videos are very intuitive

[patriciadockery]

Thank you. I was looking to prove the sum and difference identities. My book does it the same way but hearing it really helped! 

[arunrathore]

Very clear explanation of such complex identity !!!. Thanks a ton.

[sadkritx]

Thank you so much for this informative video. My books have 2 proofs ( one using a rectangle, other uses 2 triangles) but I have no idea why they used the geometric proofs since it's only limited to acute angles and also kinda hard to understand imo. They should have included these proofs, not only are they easy to understand, you can also see why the domains are (-inf, inf) instead of just acute positive angles.

[jintelcoreduoe]

great video and clear explanation like khan academy even better

[johnbarron]

Based on the sketch in the video, point S has angle A in standard position, and point R has angle A-B in standard position. The cosine of an angle in the second quadrant yields a negative ratio since cosine is defined as the adjacent side (negative), over hypotenuse (one). The cosine ratios of A and A-B both lie somewhere in the open interval (-1, 0) since cos(90)=0 and cos(180)=-1. Angles A, B, and A-B are arbitrary; no matter where you draw them, this derivation is carried out in the same way.

[opmable]

Now I understand the derivation, thanks!

[cubanoamerican]

Absolutely awesome and clear video. Thanks a lot!!!!

[sillybilly]

i also reccomend watching 3b1b’s lockdown math lecture on imaginary number fundamentals because it culminates to intuitively understanding this identity

[kaankutlu]

thanks man this is the real math dont memorize formulas

[mathboy]

Like the vid. Thanks. You should check out a program called omnidazzle. I use the bullseye feature from it when I am projecting in class and it puts a highlight around the cursor. It would really bring attention to your cursor. Just a suggestion.

[anricogideon]

Thank you very much for the explanation! Really nice and clear. One question, I noticed some question in the comment about the S and R point being on the 2nd quadrant but you didn't write the cosine to be negative. Am I correct to assume that it doesn't matter? Since most of the sum and difference identities problem will divide the one big angle to 2 acute one hence it'll only require the first quadrant. Also, the identities will still stand regardless of where you draw the line, you can even draw them all to be on the 1st quadrant. Am I correct?