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# STEP Prep Thread 2016 (Mark. II) Watch

1. why does arctan(A)+arctan(B)=arctan(A+B/1-AB) hold iff AB<1? I understand theres issues with the range of the functions, but idk why AB<1 fixes it
2. (Original post by KFazza)
why does arctan(A)+arctan(B)=arctan(A+B/1-AB) hold iff AB<1? I understand theres issues with the range of the functions, but idk why AB<1 fixes it
Let arctanA + arctanB = arctanC. Then taking tangents implies that C = (A+B)/(1-AB). This is the forward implication.

Therefore C = tan(arctanA + arctanB). Taking inverse tangents, we want arctanC = arctan ( tan (arctanA + arctanB ) ) = arctanA + arctanB. In general, arctan (tan x) = x if and only if - pi/2 < x < pi/2. So this equation holds if and only if -pi/2 < arctanA + arctanB < pi/2. Now, it is a fact that for A =/= 0, arctanA + arctan1/A = pi/2. Since arctan is strictly increasing, we require B < 1/A to keep arctanA + arctanB within our range.

Indeed, if A is positive, the condition AB < 1 suffices, because arctanA + arctanB > -pi/2. Likewise if B is positive. We now consider A,B both negative. arctan is odd, so arctanA + arctanB = - (arctan(-A) + arctan(-B)). Note that -A and -B are positive numbers. Our range is again - pi/2 < arctan(-A) + arctan(-B) < pi/2, the same as before. The lower inequality holds true, the upper inequality is true iff -B < 1/(-A) giving us AB < 1 again.

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3. (Original post by Ecasx)
Let arctanA + arctanB = arctanC. Then taking tangents implies that C = (A+B)/(1-AB). This is the forward implication.

Therefore C = tan(arctanA + arctanB). Taking inverse tangents, we want arctanC = arctan ( tan (arctanA + arctanB ) ) = arctanA + arctanB. In general, arctan (tan x) = x if and only if - pi/2 < x < pi/2. So this equation holds if and only if -pi/2 < arctanA + arctanB < pi/2. Now, it is a fact that for A =/= 0, arctanA + arctan1/A = pi/2. Since arctan is strictly increasing, we require B < 1/A to keep arctanA + arctanB within our range.

Indeed, if A is positive, the condition AB < 1 suffices, because arctanA + arctanB > -pi/2. Likewise if B is positive. We now consider A,B both negative. arctan is odd, so arctanA + arctanB = - (arctan(-A) + arctan(-B)). Note that -A and -B are positive numbers. Our range is again - pi/2 < arctan(-A) + arctan(-B) < pi/2, the same as before. The lower inequality holds true, the upper inequality is true iff -B < 1/(-A) giving us AB < 1 again.

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It should be noted that this is only true for A>0. It's if A<0.
4. (Original post by Ecasx)
Let arctanA + arctanB = arctanC. Then taking tangents implies that C = (A+B)/(1-AB). This is the forward implication.

Therefore C = tan(arctanA + arctanB). Taking inverse tangents, we want arctanC = arctan ( tan (arctanA + arctanB ) ) = arctanA + arctanB. In general, arctan (tan x) = x if and only if - pi/2 < x < pi/2. So this equation holds if and only if -pi/2 < arctanA + arctanB < pi/2. Now, it is a fact that for A =/= 0, arctanA + arctan1/A = pi/2. Since arctan is strictly increasing, we require B < 1/A to keep arctanA + arctanB within our range.

Indeed, if A is positive, the condition AB < 1 suffices, because arctanA + arctanB > -pi/2. Likewise if B is positive. We now consider A,B both negative. arctan is odd, so arctanA + arctanB = - (arctan(-A) + arctan(-B)). Note that -A and -B are positive numbers. Our range is again - pi/2 < arctan(-A) + arctan(-B) < pi/2, the same as before. The lower inequality holds true, the upper inequality is true iff -B < 1/(-A) giving us AB < 1 again.

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Interesting, thanks!
5. (Original post by IrrationalRoot)
It should be noted that this is only true for A>0. It's if A<0.
Ah yes, thank you. Luckily this doesn't affect my reasoning when taking A or B > 0, and in the case A,B < 0 I have implicitly used this fact.

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6. Is there actually any worth in doing the most recent papers, or would it be better use of time to say do 2000-2005 instead of 2010-1015? I don't expect them to be repeating any questions from the last 5 years, but I can also guess that newer papers may have a different style?
7. (Original post by ComputerMaths97)
Is there actually any worth in doing the most recent papers, or would it be better use of time to say do 2000-2005 instead of 2010-1015? I don't expect them to be repeating any questions from the last 5 years, but I can also guess that newer papers may have a different style?
I wouldn't base your practice on the chance of them repeating questions.
It is certainly better to do the newest papers, they are different in some ways.
8. (Original post by ComputerMaths97)
Is there actually any worth in doing the most recent papers, or would it be better use of time to say do 2000-2005 instead of 2010-1015? I don't expect them to be repeating any questions from the last 5 years, but I can also guess that newer papers may have a different style?
preparing for STEP isn't like a level where questions repeat every year it's a practicing for any question to come up because you have problem solving skills
9. (Original post by IrrationalRoot)
I wouldn't base your practice on the chance of them repeating questions.
It is certainly better to do the newest papers, they are different in some ways.
(Original post by Number Nine)
preparing for STEP isn't like a level where questions repeat every year it's a practicing for any question to come up because you have problem solving skills
It's hard for me to get motivation when I'm only building up general skills as opposed to directly learning something I'll be tested on. That's gotta be the toughest thing about STEP revision for me, finding the motivation when there's a good chance what I'm doing at any one second won't help at all. It's a much more lengthy process, progress is slow etc. Especially when you're used to how "rewarding" GCSE/ A Level revision is, in terms of learning what you'll actually be tested on, STEP takes it's toll there. I understand though, guess I'll do the more recent ones. Thanks guys!
10. (Original post by ComputerMaths97)
It's hard for me to get motivation when I'm only building up general skills as opposed to directly learning something I'll be tested on. That's gotta be the toughest thing about STEP revision for me, finding the motivation when there's a good chance what I'm doing at any one second won't help at all. It's a much more lengthy process, progress is slow etc. Especially when you're used to how "rewarding" GCSE/ A Level revision is, in terms of learning what you'll actually be tested on, STEP takes it's toll there. I understand though, guess I'll do the more recent ones. Thanks guys!
not long to go now and youll be at cambridge is that not motivation enough
11. (Original post by Number Nine)
not long to go now and youll be at cambridge is that not motivation enough
Lool nah I already know I'll not be able to hit my offer (but might still get in as it's a comp sci offer w/ maths) but I want to at least do decent out of pride. And I'm not going to lie STEP is way more enjoyable to me than doing a C3 or C4 past paper, so I just take certain questions for that too. STEP isn't too bad, it's just incredibly difficult.
12. (Original post by ComputerMaths97)
It's hard for me to get motivation when I'm only building up general skills as opposed to directly learning something I'll be tested on. That's gotta be the toughest thing about STEP revision for me, finding the motivation when there's a good chance what I'm doing at any one second won't help at all. It's a much more lengthy process, progress is slow etc. Especially when you're used to how "rewarding" GCSE/ A Level revision is, in terms of learning what you'll actually be tested on, STEP takes it's toll there. I understand though, guess I'll do the more recent ones. Thanks guys!
Yep what can happen with STEP is that you do really well in practice but still flop your actual exams (generally unlike A-Levels etc.). By practising you're increasing your chances of doing well but there's still no guarantee since the examiners try hard to make the questions original (I know there are a few striking exceptions lol) every year, it could just be a bad paper for you and you don't see the solutions etc.

The most important thing is not that you have done a lot of questions so that 'there's a chance one of them might come up' but that you've gained a good understanding of all the content and become very good at problem solving and can apply these skills to unfamiliar questions.
13. (Original post by IrrationalRoot)
Yep what can happen with STEP is that you do really well in practice but still flop your actual exams (generally unlike A-Levels etc.). By practising you're increasing your chances of doing well but there's still no guarantee since the examiners try hard to make the questions original (I know there are a few striking exceptions lol) every year, it could just be a bad paper for you and you don't see the solutions etc.

The most important thing is not that you have done a lot of questions so that 'there's a chance one of them might come up' but that you've gained a good understanding of all the content and become very good at problem solving and can apply these skills to unfamiliar questions.
Fair play mate, good points you make. Yeah my results are too scattered in STEP, one day I'll get a 1 the next I'll get a 3 not even going to lie it's infruriating. At this point I just want 2's, because although I'd miss my offer I wouldn't consider that a failure if my results were at least 2,2,2
14. (Original post by ComputerMaths97)
Fair play mate, good points you make. Yeah my results are too scattered in STEP, one day I'll get a 1 the next I'll get a 3 not even going to lie it's infruriating. At this point I just want 2's, because although I'd miss my offer I wouldn't consider that a failure if my results were at least 2,2,2
15. (Original post by Number Nine)
preparing for STEP isn't like a level where questions repeat every year it's a practicing for any question to come up because you have problem solving skills
Questions don't repeat, but I think there are lots of techniques that come up repeatedly, things like solving series of the form nx^n by viewing them as derivatives in pure, or tree diagrams loop round back to the start in stats.
16. (Original post by TheWiseMan)
Cam want A*A*A + 1,2 in STEP II and III for Comp Sci w/ Maths
Then UCL A*AA just incase, like what's almost happened, I put way too much effort into STEP and get nowhere, holding back my A level grades.
17. (Original post by ComputerMaths97)
Cam want A*A*A + 1,2 in STEP II and III
Then UCL A*AA just incase, like what's almost happened, I put way too much effort into STEP and get nowhere, holding back my A level grades.
Cambridge offer is quite hard just for compsci w/maths

My offers are much easier:
Imperial: A*AA + 2 in STEP 1
UCL: A*AA
I'm sitting all three STEP papers because I might want to change to JMC
18. (Original post by TheWiseMan)
Cambridge offer is quite hard just for compsci w/maths

My offers are much easier:
Imperial: A*AA + 2 in STEP 1
UCL: A*AA
I'm sitting all three STEP papers because I might want to change to JMC
I applied to JMC, looked like an amazing course.

But because Imperial think they're way better than they are, they gave me an A*A*AA offer, meaning I had to reject it. They really are a bit silly there.
19. Everytime I look at a STEP solution my first reaction is always "Well how on earth did you decide to do that". STEP is 99% intuition.
20. (Original post by ComputerMaths97)
Everytime I look at a STEP solution my first reaction is always "Well how on earth did you decide to do that". STEP is 99% intuition.
It helps doing all the papers.

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