Maths videos to help you improve your GCSE and A-level results.
Over time I have found that I like to try to answer the "why" questions, not just the "how" questions. Why do we do it this way and not that way? If you are looking for how to learn certain skills in mathematics, you will find that here. If you are also looking for the "why" of things, you will find a decent amount to interest you.
Hello! Thank you for your very clear explanation, i have a question: this proofs seem true only when alfa and beta are acute angles. How to prove that they hold for any alpha and beta (even negative or 188 degrees or 520 degree or any other angle)?
Great question. These kinds of geometric proof don’t prove the case for angles > 180 directly but you can use other identities like sinx = sin (180-x). Eg if you have sin(150-x) it’s equivalent to sin(30+x) or sin(230-x)=sin(-50+x). I think that should be enough to extend these proofs.
@@mathonify Thank you for the answer, but at that point one could ask: "ok but how to prove the other identities?" I think the true starting point (of the other identities too) is taking the distance between any two point on the goniometric circumference and proving cos(x+y) = cos x cos y - sin x sin y. Then from there gaining all other trig formulas whatsoever
I think this trigonometric eq could be solved another way: if we multiply both sides by sin(x), and use the identity 2sin(u)cos(u)=sin(2u) twice, we reach this equation: sin(4x)cos(3x)=sin(x) Now, using the identity sin(p)cos(q)=1/2(sin(p+q)+sin(p-q)), we reach 1/2((sin(7x)+sin(x))=sinx or equivalently, sin(7x)=sin(x) which results in 7x=2kpi+x and 7x=(2k+1)pi-x which leads to two general answers: x=k(pi/3) and x=(2k+1)pi/8 Since k*pi which is the zero-making roots of the multiplying factor, sin(x), and is not the root of the original equation, we have to put this limitation on k: k can be any integer number but the coefficients of 3. For the 2nd general answer, we don't have such a problem insomuch as 2k+1 would never be equal to any integer coefficient of 8 and this form of answer is valid for any k.
You could have done Q14 in an easier way We know that the ratio of tulips to roses is 6:5 so we can say 6/11 of the total number of flowers are tulips We also know that the ratio of red tulips to yellow tulips is 3:4 so we can say 3/7 of the total number of tulips are red So out of our total number tulips (6/11) 3/7 of them are red So we do 6/11 x 3/7 which will give us 18/77 And the video was great! Thanks for the help!
