The students in that class were so fortunate to have Eddie Woo as their teacher. This guy is something else, after just 7 mins I have solved my first Trig identity proof.
I wrote the incorrect steps in the exam and got marked wrong for it. I asked my teacher why and she don't give me a reason for it. I had argued with her in front of the class for half the lesson and still don't get why. And here Mr Woo just explains it in literally 2 mins. I thank you so so much for the explanation!
In short, you're using the statement that you've been asked to prove to prove the statement, and since that relies on the statement being true in the first place, it's a logical fallacy known as circular reasoning.
Perfect! I have seen this a lot from my students who go to my class with prior knowledge in trigonometry and they present their proofs by starting with the "equation" instead of working from one side to reach the other.
This is exactly how I prove trig. identities even before watching this video. I make an end for both sides through simplification and manipulating identities then makes those ends meet. I learned a lot through this video resurfacing my knowledge about proving trig. identities
I wish this guy was my math teacher. like the energy and hand gestures and the way how you simplify your explanations so its so easy to understand.... Man thank you so much. Its my first time watching your video and I was hella sleepy, but I felt like I was in the actual classroom when I was watching this video.
There is a simillar problem with prove by induction. By plugging in n + 1 into a formula and ending with 0 = 0, you just prove that zero is equal to zero. Exactly as you say, you need to end with n + 1 equivallent of the original formula.
If each equation is known to be equivalent to the previous one (i.e. you have a string of if and only if's) then there is nothing "wrong" with the first "incorrect" proof you gave. You are simply showing that Equation 1 Equation 2 Equation 3 and since you know Equation 3 is an identity, the first equation must also be an identity. It's a perfectly correct approach to take, provided you understand the reasoning involved. You're not "assuming what you're trying to prove." You're proving that the identity you want to prove is equivalent to a known identity. What you CAN'T do is start with an equation you want to prove is an identity, prove that this equation implies an identity, and then say you've proved the original equation is an identity. For example, you can't start with -3 = 3, square both sides, and say since 9 = 9, we know that -3 = 3 is an identity. That's because the implication only runs in one direction. I would also say it's incorrect to say that an identity is NOT an equation. An identity is simply an equation that is true for all values of the variables under consideration.
I'd say this: all you have to do is rewrite all the same formulas in reverse order (starting from the identity cos^2 theta = cos^2 theta, and ending with the thing you're trying to prove.) You ALSO need to check that each logical step is still valid (no multiplications by zeros, or multiplications by *anything* not known to definitely be nonzero), and you're left with a rigorous proof.
I agree with you, Darin. The danger of solving an equality as if it were an equation (as in the example Eddie marks as "incorrect") is that some students will inevitably get tempted to switch terms between the left- and the right-hand side. In the example given, that would mean moving the cos^2 to the left and the -1 to the right to obtain the known Pythagorean identity, and then concluding that the original identity was true because the Pythagorean identity is true. And that would not be a proof.
2:28 - 55 “I started with this, I can’t start with this I’m supposed to end with this so that can’t be the first line that’s gotta be the last line I’ve got it completely backwards. You’re meant to prove this the burden of proof is in you it’s like walking into a legal court and you’re the defense well everyone assuming THAT my client is innocent... you can’t use that as your basis”
If it says verify the first option he showed is valid. Since it says "prove" you cannot use what you started with in your proof. Usually these type of questions are written a little different to avoid this issue like "given (1-sinx) and (1+sinx), prove that their product is equal to cos^2(x). Then you dont even need LHS and RHS. Just do the distributive property, use the identity and end proof.
Its not hijacked, the topic is actually proving whether these identities are equal on both sides. The topic you are referring to should be proving trignometric functions
I still don't understand why you can't just start with the identity. If you start with the identity, and then compute the equation (via algebraic manipulation/known theorems) until you reach 0=0, you've effectively proven that the identity is just as true as 0=0. Since 0=0 is clearly true, then so is the identity, and I believe that you have provided sufficient proof. If you were to do this with something that wasn't actually an identity, your final line of working would not reach something that wasn't unequivocally true (i.e. you wouldn't be able to get to 0=0). Therefore this line of working is something that's non-replicable with non-identites, and we can accept it as proof for the given identity. Can someone please help me understand what is wrong with this line of thinking? Thank you.
But there is literally nothing wrong with the first method. Just use bivariate implication arrows. Also if LHS = sin^2 (t) and RHS = sin^2 (t) then LHS = RHS by transitivity of equality - no need to write ANYTHING in reverse. . There is absolutely NO NEED to manipulate the LHS into the RHS or vice versa - I wish teachers would stop claiming this as it stifles creativity when constructing proofs. A proof by contradiction also works. Let's assume that there exists a value of t such that the inequality doesn't hold - then you can then argue from that, that cos^2 (t) doesn't equal cos^2 (t).
@@tastypencil But what you're saying isn't actually the exact same thing. It's right, but when we're trying to prove an identity like, say, a = c, we can't assume that a = c is true. In other words, if we had to prove it, our steps shouldn't start with a = c, because that's exactly what we have to prove. So what you have to do is take LHS, a, the do some nifty maths, and make sure you end up with c. So, taking your example of: Prove: a = c a = b c = b Hence, a = c. Sure, it's mathematically correct. But where did you get c from? The questions is asking you to arrive at c from a. Not take both a and c, and arrive at the same thing, b. Your first step a = b is correct. But you have to work in reverse to take b and end up with c. Basically, you must end at c to successfully prove the identity. Bit of a long answer but I hope that helps.
@@aakashm9962 i mean ye but commutative law? if it asks to prove a = c and u start with c and prove it equals a, youve proved c=a, which is the same as a=c The only thing you've assumed is that c exists. I get what you're saying tho, it appears to be just a convention of proofs that you have to start from the side it says and then end on the RHS. Seems dumb tho, i mean when you're literally reversing your working out Thx for the clarification!
@Aakash Mandava It makes no sense to say the "RHS is true". The RHS is an expression, not an equation. If you want to prove a = c is an identity, it's perfectly okay to prove a = b and c = b are identities and then combine them.
How to prove the the following identity 'sin8x=8sinxcosxcos2xcos4x' step by step.. I've been stuck on this one for 3 hours. Any help would be great thanks 😊
That's rather silly. In the first example, all steps use logical equality so it's a perfectly fine proof. In the last example, just finish up with LHS = sin²θ = RHS. There's no need to waste time writing stuff in reverse.
He just wrote down the exact same stuff as in the first example. Also, whenever you have to proof something is true there is no wrong in assuming the point is true; Whenever the point to proof is 'false', you will eventually encounter a statement like 1=2. I don't seem to get what's wrong with the first ex.
You are asked to prove that LHS = RHS though. To do that you need to show that starting at the LHS, you can end up at the RHS. Notice the first example have the same steps both times. But writing '= cos^2(x)' each time is bad in the sense that that is what you are wanting to end up with. Instead of proving that LHS = RHS, you are assuming LHS = RHS and then verifying it, which is not the same. Again with the second example, you are wanting to start from the LHS and after multiple steps, finish with the RHS. It may sound pedantic, but by just saying LHS = sin^2(x) = RHS, you have not shown the steps to get from sin^2(x) to RHS, so you have not yet shown how to get from the LHS to the RHS.
@@XxStuart96xX if you assume an identity is true and could manipulate it through biconditional operations to a true statement that means the original identity must be true. High school teachers may not like it, but that's totally valid in formal logic. If it were false there would be a contradiction, resulting in a false statement.
So tell me why 1-sin^2 theta =cos^2 theta. Otherwise, it's just a(n) (il) logical mathematical syllogism that has no meaning or purpose. So what is its purpose? How can this used in real life situations to solve real world problems? If it can't, I don't see the point of it.
Wtf is wrong with you. It's an if not an is. It doesn't say that it's outright true but we're going within the constraints of mathematics to check if it's true.
u know the feeling when u r trying to make it too easy for the students and then we just mess up the whole class....so the same thing written on top is wrong but on bottom is right..da hell is ur point ..
