Remembering that sin2(θ)=1/2−1/2cos(2θ)\sin^2(\theta) = 1/2 – 1/2\cos(2\theta)?

How do you remember this for integrals? It doesn’t seem obvious and I can never remember it when I come across it in integrals. Answer The double angles should be remembered as follows, 2sin2(θ)=1−cos(2θ)2cos2(θ)=1+cos(2θ). To know which sign for each, remember that ‘sinning’ is bad, hence it leads to −. Reason why you should memorize … Read more

What is the good way to remember the signs of the rotational matrix?

Recall rotational matrix in (x,y) is given by: $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ For the life of me I cannot remember if the top right sin entry has a negative or positive sign, and I have known this matrix since 5 years ago (obviously I am a failure!!). Sure … Read more

Remembering numbers while performing calculations mentally

Specific question: how can I improve on mentally performing calculations like these : $$(78-59)\times23-8\%\times(270+130)$$ $$[2\text{digit}+-2\text{digit}]\times2\text{digit}+-x\%\text{of}[3\text{digit}+-3\text{digit}]$$ Without writing anything down. The thing is I can do the first part before the second minus pretty fast $19\times23=460-23=437$. When I move on to perform the second part $8\%\times(270+130)$ and I will come up with $32$ pretty fast but … Read more

Mnemonic to remembering that inverse limits are limits and direct limits are colimits

Sometime ago I had trouble remembering that right adjoints preserve limits (and so left adjoints preserve colimits). But ever since R. Vakil suggested that we use RAPL as a mnemonic to Right Adjoint Preserve Limits, it stuck on my mind and I never forgot. However I still have trouble remembering that Inverse limits (= projective … Read more

Mnemonics for linear algebra

Sometimes formulas in linear algebra are not easy to remember. Some usefulness for the process of remembering can provide application of mnemonics. Do you know some useful mnemonics for this purpose? I’ll give two examples: For the process of finding the inverse of matrix it could be used mnemonic Detminstra what can be translated as … Read more

Are there examples of when the ILATE mnemonic for choosing factors when integrating by parts fails?

Is it possible in some cases that using the ILATE rule does not yield an explicit antiderivative but making another choice does yields one? If so, please give examples. Answer Yes, it’s easy for the rule to fail if the proposed derivative is not integrable. For example in the integral ∫x3ex2dx the rule would propose … Read more

Remembering multiplication of these two numbers: 7×8=567 \times 8 = 56 and 9×6=549 \times 6 = 54

I have almost mastered multiplication table up to 9×9 however, I’m having problems with the following two. 7 x 8 = 56 and 9 x 6 = 54 For some reason my brain thinks that 56 and 54 are somewhat the same and sometimes confuses the two multiplication problems. I was wondering if there is … Read more

Is there a reason why we call Hom(A,−)\mathrm{Hom}(A, -) covariant and Hom(−,B)\mathrm{Hom}(-,B) contravariant?

I have trouble remembering what it means to apply the covariant or contravariant Hom functor. For example, when I see Hom(A,−), I always forget if I am going to reverse arrows or not. For example: I used to have trouble remembering what a pushout and pullback was until someone pointed out that pushout diagrams have … Read more

Sine and Cosine Derivatives

The derivatives of: ddxsin(x)=cos(x) ddxcos(x)=−sin(x) I currently trying to teach a friend of mine calculus, because he does not know it yet. He keeps forgetting how to take the derivatives of sin(x) and cos(x). Is there a simple way, or trick to remember the derivatives of sin(x) and cos(x)? Answer Not that I know of. … Read more

Memorizing the identities cosπ3=sinπ6=12\cos {\pi \over 3}=\sin {\pi \over 6} = {1 \over 2}

I memorized sinπ4=cosπ4=1√2 easily by using the diagonal inside the unit square. I am having great trouble memorizing the identities cosπ3=sinπ6=12 because I keep confusing whether it is cosπ3 or cosπ6 that equals 12. Is there a picture similar to the unit square picture or something like it to memorize this identity? Answer There is … Read more